I like Dorian's answer (go upvote it!), but that was not how I read the question. I am going to answer the question exactly as it is asked: nothing more, nothing less. That is, there's a portal with the following constraints, how long would it take to rebuild on the other side? No "What is this thing?" or "Should we step through it?" No experiments. Just: portal. build fast. GO.
(WOW! This post became huge. It's sectioned well though, so I'll leave it as is. Enjoy.)
TL;DR: On the order of 1-10 years if it just pops in and enough skilled people agree to try and rush-build a new modern society there and they coordinate well. On the order of 1 month - 1 year if we know it's coming and can prepare and practice for it and have a colossal level of support and expertise from thousands or millions of people.
Disclaimer: This answer assumes the right people come together and that they all work hard and get everything done fast, possibly cutting corners along the way, even safety corners.
Other assumptions: The area on the other side will have reasonable access by foot to forests, plains, rivers, various types of resources such as iron ore and clay, and other things we have here. You might have to spend some time (hours or days) searching for these things on the other side, but they are there. I also assume that you have skilled people for each task every step of the way. They don't have to be the best of the best, as long as they have the skill. And concerning only living matter passes: I'm going to bring plants, but only if they are fully mature plants still alive enough that you could plant them in the ground and they could keep growing, that should count as alive.
This might sound unrealistic, but there are various reasons this could happen. Maybe the portal appears, the experiments Dorian mentioned were carried out but we don't count that time (or maybe not), but we're at the point where we understand the rules of the portal and we're forced to just build fast. We find out aliens are going to invade Earth sometime soon, or whatever. We have a fire under our butts.
Remember, you can send dozens or even hundreds of people through to work for the day, then they can come back to eat and sleep on Earth. Or they can carry armloads of fresh (still living) fruits, vegetables, and animals and just eat them on the other side.
#Go!
##Starting with primitive tools
Fortunately there are crazy awesome people who do similar things just for fun or research who we can use as guidelines. Go check out the YouTube channel Primitive Technology if you haven't already done so.
Some of what follows is also from personal experience. I consider myself to be a hobby bush-crafter, or hobby "primitive survival" practitioner. I've made stone tools, started fires with flint and steel and with friction (ie: "rubbing sticks together"), made rope from plants, fired pottery the old fashioned way, dug garden beds with a stick, and done other things you typically associate with this type of activity.
###Hand Tools
As long as you have various kinds of rocks you can make some very crude stone tools useful for a few tasks in less than 1 hour. If you get experienced flint nappers in your group who find flint on day 1, you could have sharp knives and axes on day 2. Also, if we are allowed to bring an animal across such as a beaver and then extract its tooth, those can make good carving tools so you could have one almost immediately, but I'll just assume you wait a day for the good flint tools. By the way, flint tools can easily be made sharper than iron tools, they just have other drawbacks.
###Clothing
If you have sharp flint knives on day 2, then bring the animals over and make clothes out of the hides. If you skin it and wash it you can start wearing it almost immediately. It might seem gross, but I'd rather be wearing a hide that still smells like its previous owner than to keep going naked. Clothing partially accounted for day 2.
###Fire
With good fire starting materials, even natural ones, the best fire starters can walk out into the wilderness, grab what they need, and have a fire on day 1 if there are some very dry materials available. It could take a few days or weeks if the materials need some processing such as drying in the sun or carving with the stone tools.
###Manufacturing
Very primitive manufacturing was done by shaping wood or making pottery. With the right people, you'll be chopping down wood and carving it on your very first day here. Pottery will take longer since you have to find a good clay source, shape it, dry it, then fire it.
Wood manufacturing will start on day 1 or 2. Clay firing will start after a few days for small items if you have fire by then, up to a few weeks for larger items that take longer for the clay to dry.
###Farming
Someone can start working on a garden bed immediately on day 1. Even if you want to tear up the ground (not necessary for all farming techniques) you can still do it with nothing more than a stick you pick up or break from a tree.
###Build Time: Primitive Technology
After 1 or 2 weeks you will have:
- huts for shelter, a shed full of knives, axes, adzes, rope, and other simple tools.
- a garden bed full of plants transplanted from Earth
- whatever they want made of clay including plates, bowls, bricks, tiles, etc.
- stone hearths and fireplaces
##Metal Age
They will have already started on the building blocks for more advanced technologies before they complete the "primitive" section above. Someone will already have started working on a clay furnace and forge. Someone else will already have started working on a bellows. Others have already been searching for ores. I'm just calling it the metal age since they can start on them all at the same time, even iron. And I'm going to concentrate on iron here.
As soon as 1 or 2 furnaces are ready, someone will already have found at least a few rocks with ores in them. Even if it takes a few tries, there will be smelted metal in less than a month, especially if a bellows is completed in time.
The first metalworking tools are sticks and stones, so the first tools made out of metal will have lots of impurities and be very crudely made, but they can function. The first iron pokers, hammers, and knives on wooden handles will be hastily made all still inside that first month.
Using those tools, better tools will be made before the second month is up. These tools will include all of the conventional blacksmithing tools and some knives, axes, adzes, saws, horseshoes, barrel rings, wheel bindings.
Yes, even horseshoes and wheel bindings: we might as well get the beasts of burden over here now if they aren't already there. Make sure you've been working on carts and carriages in the meantime.
After just a couple months you have homes, gardens, metal hand tools, flocks of animals, vehicles, barrels, and lots of other things. Since the heating technology is already well advanced at this point you could make glass too as long as you've acquired the materials for it.
##Manufacturing Age
At the end of the previous section, you've already had everything needed to start making the first woodworking lathes. Most of it can be made out of wood, and the cutting edge can be as simple as a blade or a sharp point. I have done rough lathe work before by putting a stick in a drill, attaching something to the stick to spin it, and applying a knife held in my other hand to the wood. For our group, they don't have the hand drill, but they could make something very similar.
The first woodworking lathes could come before you even have your iron work going since you could do it with flint cutting tools. But once you have iron tools it will be even more reliable.
By the end of the third month setting up this world you will have lots of lathed wood objects, and as soon as a blacksmith makes an iron drill bit you can get a woodworking mill going too, so your manufacturing base has essentially started.
What you can not do well at this time is mill or lathe iron objects, but you can use iron mill and lathe cutting tools on some other metals like copper, tin, or lead. And you'll have various other tools at this point for rolling metal, bending metal, etc..
By the end of the third month, your metal industry is mostly limited by how much metal can be acquired. If you can get lots of it, this world will have transformed into something that looks like Earth in the 17-or-1800s. Lots of things could be mass manufactured, even non-iron metal objects.
At this time iron objects, including steel which is made from iron, will probably still be made on an anvil.
##Chemistry
So we're three months in and we have all this cool manufacturing, blacksmithing, glass, etc., but to continue making this colony into something resembling modern Earth we need chemistry.
We're not starting on the chemistry just now, instead the work should already have been ongoing since about the second week on this world. Ever since we had fires, primitive chemists will have been starting to create lots of chemicals. Once there were clay vials, they can be working on liquid chemicals too.
Before the end of the previous section, there will already have been gasifiers, beakers, tubes, and various other tools created and we'll be at a chemistry level of technology similar to the 1800s or early 1900s, except that we'll only have relatively small quantities of the chemicals so there will be an effort to simply mass produce as much as we can before we can actually start using it to do much of interest.
Two of the more useful things that might come out of this are fuel and aluminum. Fuel might come from gasified wood or from plant ethanol. Aluminum will be made using the Bayer/Hall-Heroult Processes.
Aluminum is important because it has a much lower melting point than iron, it makes a fine conductor (though not as good as copper), and it can be worked on easier than steal. Also the ground is full of tons of the ingredients for aluminum.
Another big one to try and push for soon would be plastics and rubbers. Then we can start making covers for wiring to get ready for our later electronics.
With our manufacturing in full swing, aluminum available to compliment our blacksmithing steal work, and fuel from plants, we might be able to start working on some automotive prototypes, the first ones probably wood frames with the simplest engine we can make.
A lot of the chemistry initial buildup will happen at the same time as the previous sections, but some will have to wait until we're manufacturing stuff. We might have some prototypes for some of these things likes automotives and wiring by the end of the 4th or 5th month, but we would have to create a factory before we could mass produce these things. The initial versions will be crude and potentially dangerous, but they could be rushed into existence around this time.
##Electricity
It might sound like a bit of a stretch, but it would not be unreasonable to start generating some electricity as soon as we have conductors that we can start connecting fruits and vegetables with. Most people have heard of using citrus fruits as power sources, but you can do it with other thing too such as potatoes. As long as you bring a still-live one over, so it goes through the portal successfully, you can use it as soon as you can wire them up. In some stores you can even buy kits that power things this way. My kids had a potato-powered clock kit at one time.
Barring that we can start making batteries if we find the necessary materials. If you have the correct materials at hand, you can make a battery just by piling the right materials together in the right way.
Both of the previous two electricity-generating methods could be done as soon as you can make the wires, so within months of arriving.
As soon as we have wires and magnets we can start making electricity the way we actually do it in power plants. Once we get to this point, we can set up larger scale power generating plants. We could do this within months of arriving as long as we have magnets. To actually do much of interest with the electricity will require the manufacturing from the previous section.
##1900s level
We're only a half-year in and we're already building devices to use with our electrical grid and we have manufacturing and chemistry. How long it takes to have various useful devices depends on their complexity.
In the first year some examples of things you could expect rough versions of:
- electric motors/engines
- lights
- heaters, stoves
- radios
- air-tight containers
- airplane prototypes (early Wright brothers level)
- 1800s style "submarines" (they are not what you imagine when you say submarine)
- rocket prototypes
- 1800s or early 1900s style automotives
- pumps (both water and air)
- telegraphs and maybe telephones
In the first year you should not expect:
- electronic computers, at least not useful ones
- TVs, monitors
- useful rockets
We might be able to have some form of crude 8-bit electronic computer in the second year. It would probably be just a few of them built, they would be designed for their specific purpose, and they would be used only to help bootstrap the rest of the computer revolution. These initial ones would be slow, mouse-less, monitor-less things whose time would be very valuable until they were used to help design and create the next generation of computers. Remember, this is being assisted directly by Earth computers and engineers on the other side of the portal, we only need the ones here just to get some form of automation going and then that world's computing revolution can explode.
Since we are assuming limited previous life on this planet there likely will be no major source of fossil fuels. That is just as well: it may speed the development along, but it's all the better that we start right out developing renewable, sustainable, cleaner energy sources. It would probably start with wind and water the first year, and we might be trying to make solar panels the second year.
So I think we would see prototypes and some more functional thing we associate with 1900s level technology before the end of the second year.
##2000s level
We can get up to 1800s and some early 1900s level by cutting a lot of corners. But going that last step to having very precise machining and manufacturing of precision parts and having microscopic-scale manufacturing for computer parts is going to be one of the trickier parts.
How long this would take is a hot topic of debate among some people. I feel the research is inconclusive.
It certainly should not take decades though. It took decades on Earth. If we have the push for this that I described initially, we should be able to do it many times faster than it took us on Earth. After all, we already have it and understand it, we're just recreating it, and we're doing so with all experts readily available.
So going from the end of the previous section to being fully 2000's style modern should take on the order of years. I would say months, but there are activities which need to be physically done, tested, possibly redone, such as physically creating computers, rockets, planes, etc.. It's not just a matter of drawing the design up in your first computer you make.
Conclusion
All of the initial setup checkpoints will take days, weeks, or months. Then to get to the level of a few hundred years ago will take months, setting up mass-production (relative to what is already there) will take months. After a year or two things will be similar to Earth a century or two ago.
Then progress will slow down as you approach modern Earth level and you try to make everything very precise, very large scale mass production, build up a large quantity of essential components, and recreate complicated structures like modern rockets, modern cars and planes, and modern computers, medical equipment, etc..
The entire process from day 1 to the day you could almost mistake it for Earth today would be on the order of years I think, not decades, and not months just because it takes time to perform the activities.
##Bonus Round!
Now I'll take it a step further. Let's assume we've made contact with ETs and befriended some alien races. They find out about our Minecraft game and they love it! They create "Minecraft Live!" which is exactly what you described in your question and it was designed and created by the most advanced of the ET races.
Each time the game is started the portal shows up on the participating race's home-worlds. You have advance notice about when and where the portal will appear, and you have time to prepare.
In this case, you are able to form a game plan and have a line of people waiting at the portal location, all the people you need to enter that day. They all know exactly what their jobs are and what they plan to do that day (or longer if they plan on staying more than a day). As soon as the portal appears, the single-file line can rush in and everyone can hurry to their required activities. It is the largest-scale multiplayer game in the history of the universe.
Also people will be working 24-7, all day every day. Not individual people, but they will work in shifts. And this is the second time humans have joined this game, and they have refined a lot of their process after doing a previous round of the game.
###Day 1
Thousands, maybe tens of thousands, of people pour through the portal. Lots of primitive tool makers, fire starters, resource gatherers, gardeners, etc.. The last 10% of the line of people are carrying armloads of live plants and small live trees and bringing animals with them.
Primitive tool makers and their resource gatherers run to find resources and have the first tools made within minutes, then the first reasonably sharp ones made within hours.
Fire starters find the best resources and work together in teams using the hand-drill friction fire starting technique. They've practiced a variant just for this game where they work together, switching places every half minute so they can exert as much energy as possible even though that tires them quickly, because fresh bodies are available. At least one of the teams manages to start a fire before nightfall.
Chemists already have armloads of reagents, some are already performing their chemistry with formulas that don't need fire. The ones that do need fire are anxiously waiting to pounce on a fire as soon as it's ready and start making various chemicals.
Potters are making containers, but not for dining use: we're concentrating only on what drives the game as fast as possible. They're working for the chemists at this point to get them chemistry tools. Others are already molding furnaces and forges. By the end of the day it's all made. They will set them near fires; not close enough for firing because it's too early for that, just close enough to dry for now, then fire them later.
Woodworkers are making all manner of things. The first sharp tools were made within hours by the flint nappers, so they are carving and chopping everything: huts, lathe frames, vehicle frames, looms, everything you can think of they are already starting on even if it's not needed yet.
###Day 2
People are already starting to try to rush certain things. Sometimes iron work can be done with straight wood instead of charcoal or coal, so people are trying even though it might not work. If we can smelt some iron before the first iron furnaces are ready, that would be a big leg up.
There are also horse-drawn carts and carriages already. Extremely crude, and they might even break down frequently, but they are there. So we can already start on lots of heavy lifting.
The first lathes are ready with flint cutting edges and points so lathe-based wood manufacturing can start before the end of day 2! Remember, everyone is helping and working hard, and multiple people coordinated and practiced ahead of time to work together building this lathe, so this is not unreasonable.
###Day 3
We have some metal ingots poured into wooden molds. Maybe even some smelted iron if any of yesterday's iron corner-cutters were successful.
The lathes are being used to make more lathes and to make mill frames which will receive copper or tin drill bits tomorrow. We are sacrificing a little bit of progress on day 3 to make more of what we have to drastically speed up progress on later days.
Chemists are already making tons of different chemicals and utilizing the first cups and beakers and clay tubes the potters are making.
###Day 4
Large clay objects like furnaces and forges will take much longer to dry, but we want to rush this process so we start firing them up today even though they haven't fully dried. Most will break, but we should have a few that survive being fired up even after only a few days of drying. We made lots and are hoping a few make it. So we can get started with iron working before the end of day 4.
Workers of other metals are already starting to make the first electrical conductors. If we can make copper or aluminum by now, they'll use that. If not, they'll make do with whatever they can; there has to be something they can make or use by now. Others have been stringing together potatoes into a potato battery, or maybe the resource scavengers have built up reasonable batteries by now. Either way, we're making wires and we have the first crude electricity sources.
###End of Week 1
By the end of the first week our feverish scurrying mass of The Ultimate Workforce (Don't tell them that though! Remember, it's a game, and that's why they are all trying so hard!) has gotten a lot of chemistry, manufacturing, metal working including iron and aluminum, and wires and batteries going. They have also hurried out the first motors to use with the fuels the chemists have managed to create so far.
The first motors will be dangerous, but surely there will be people who will operate them anyway. Some will explode and there might be some injuries, maybe even a death or two.
Others have started working on the first iterations of other advanced technologies: rockets, computers, vehicles, etc.. Even if they don't have enough to finish them completely just yet they are still getting done what they can. A week in and we're already trying to rush out crude forms of 17-1800s level tech, and maybe a few early 1900s level things.
###Finish Line
Incredibly, we managed to get to 1900s level tech on lots of things by the middle of the second week. Now you want the rocket-to-mars and nano-scale-computer level of manufacturing precision. You have hundreds of experts in every field all lined up, maybe thousands for computer science. They can skip over a lot of the "Let's build a computer to help crunch the numbers for the next computer" type of iteration, because a lot of them have memorized portions of computer hardware schematics or portions of an operating system. As soon as the first thing are built that can accept data and use it to help craft the next gen, the 100 keyboards that were created just for this are all connected in and the engineers all do a massive data dump of what they memorized.
This does require a reasonably sized data storage technique. If one cannot be fashioned that fast, perhaps the computer is not fully automated but instead there is a bank of humans trained to act as a computer and they have a line of people who feeds input to them and they do the job of playing hard drive and possibly even CPU. Something could be worked out, and this human-computer-network can help manufacturer the first useful computer and all its components.
And also in the meantime all the other advances are being made, but people usually care about metallurgy, space travel, and computers when they ask these questions.
All this will probably take on the order of weeks to months. That on top of the initial 2 weeks means that maybe you could get up to current level in somewhere between 1 month to 1 year.
##Minecraft Live! Conclusion
This is the best case scenario for fastest time, and it is borderline crazy but might technically be possible, especially since there is advance notice and preparation, and maybe even practice runs.
Now I feel like I want to start a club who actually does this type of thing. What government will let us do all this in their border?