Summary
What you're trying to do doesn't make sense unless you send a few thousand (or more, depending on sensitivity) mapping drones into the galaxy to keep up-to-date mapping data. The drones would be connected in an FTL sneakernet so you can just ping the nearest drones for data from time to time and have pretty accurate data.
Note that if you're in the thousands of probes range, the data will necessarily be predictions from light thousands of years old. The simulations should be very accurate with advanced math, but things like "the Death Star exploded your planet" won't show up.
Additionally, without extremely good resolution on the drones (not likely due to diffraction limits), there will always be uncertainty about the exact positions of small bodies in the unexplored / infrequently visited solar systems. As such, you'll need to send a scout drone ahead to be absolutely certain there's nothing where you're going in you're in unexplored areas.
However, mapping within explored solar systems would be done routinely so there will be no issues there.
You're pretty safe blindly jumping anywhere you can see, so it's not that huge of a deal anyways1.
If you want one-year "real-time" accuracy, you'll need hundreds of millions to hundreds of billions of probes. Same thing if you want to see all the planets in the system without probes jumping ahead.
I didn't do any math, but the entire network could potentially be setup within days if your mapping drones can jump hundreds of light years at a time. However, it will take some time to calculate precise velocity information to make your models accurate. A few years should be fine, but an advanced civilization could probably do it faster. From there, updates every few decades should be sufficient.
Can't take a snapshot from a single point.
There's no way to take a snapshot of the entire galaxy from a single point in the galaxy. The galaxy itself blocks light from reaching us, plus there's a resolution limit due to diffraction. There's (probably) also the pesky super-massive black hole in the center of the galaxy that's difficult to see past.
That said, you could get pretty good pictures of the galaxy by using a number of probes spread through the galaxy. There's no way you'd get exact positions of all the planets, asteroids, and smaller junk from really long distances, but you could at least know where all the major stuff is.
Highly populated areas would have lots of data available for real-time updates on pretty much anything hazardous. Unexplored areas would require mapping before a jump.
Need lots of probes. How many?
Future technology will improve our resolution to an extent, but let's say 1000 light years is a good range to see most stuff (this page says we can use parallax out to a couple thousand, but let's be conservative). Then we just need to cover the galaxy in a 1000 light year grid.
The galaxy is roughly cylindrical. From Wikipedia, it's about 100k light years in diameter, and 1k light years thick. That's a volume of about $2\pi rh$ $=2\pi(50kly)^2\cdot1kly$ $\approx6\cdot10^{13}ly^3$.
In 3D space, the furthest points in a rectangular grid are $\sqrt{d^2+d^2+d^2}$ $=\sqrt{3d^2}$ $=d\sqrt{3}$. The furthest point should be in the middle of a grid cube. This means our grid needs to be $2\cdot\frac{1000ly}{\sqrt{3}}$ $\approx 1155 ly$ from node to node in order for every star to be within 1000 light years of a station.
Originally, I thought of a bunch of cubic groups. Each group will have eight stations (the corners of the cube), but almost all of those stations will be shared by four or eight cubes. Each cube will be $(1155ly)^3$ $=1.54\cdot10^9ly^3$ in volume. That means we need about $\frac{6\cdot10^{13}ly^3}{1.54\cdot10^9ly^3}$ $\approx 39000$ cubes. Since each cube has eight stations, and most nodes are shared by eight cubes, that's around 39000 stations. Note, however, that this only applies if you're putting a lot of stations much closer together.
It gets a little tricky here. In a 3D grid, each station will be shared by eight cubes. So there's approximately 1 station per cube, as above (technically, the outer edges will have un-shared stations, but they're in the minority). But once your view distance is over 1000-2000 ly (the thickness of the galaxy), we just have a single layer covering everything, so it's a 2D grid. In this configuration, each station is shared by about 4 cubes, so there are 2 stations per cube.
The blue dot is a star at the edge of the galaxy. It's maximal distance to a station given a single layer of cubes is given by the green vectors, which are equivalent to summing the maroon, red, and orange vectors. The distance is $\sqrt{m^2+r^2+o^2}$. The maroon and red vectors are both half the distance between stations, or $m=r=\frac{a}{\sqrt{3}}$, and we need the total distance to be less than $a$ (where $a$ is the maximum distance we need). Plug that all in to get:
$a=\sqrt{2\cdot\frac{a^2}{3}+o^2}$
$a^2=\frac{2a^2}{3}+o^2$
$o^2=a^2-\frac{2}{3}a^2$
$o=\frac{a}{\sqrt{3}}$
Total thickness is $2o$ (same distance above and below the cube) plus the height of the cube, $\frac{2a}{\sqrt{3}}$. If $h$ is the galaxy thickness, then:
$h=2\frac{a}{\sqrt{3}}+\frac{2a}{\sqrt{3}}$ $=\frac{4a}{\sqrt{3}}$
$a=\frac{h\sqrt{3}}{4}$ $=433ly$
So one layer of cubes will blanket most of the galaxy (the central regions are a bit thicker) if you're putting enough stations to keep one within 433 light years of every star. In this case, we're no longer comparing volumes, but areas. The galaxy is $\pi r^2$ $=\pi(100000ly)^2$ $=3.14\cdot10^{10}ly^2$ in area. Each cube on the grid is $(\frac{2a}{\sqrt{3}})^2$ $=\frac{4a^2}{3}$. For $a=1000ly$, we need $23550$ cubes. Each station only shares about four nodes, so you need twice as many stations as cubes, $47100$ stations total.
Additionally, if our stations can see far enough, we don't even need cubes. At that point, we just need a 2D plane of stations. We can use the same math as above, except use a zero-height cube.
$h=2\frac{a}{\sqrt{3}}+0$ $=\frac{2a}{\sqrt{3}}$
$a=\frac{h\sqrt{3}}{4}$ $=866ly$
Again, we're comparing areas. Each square has four stations, and most stations are touching four squares, so it's one-to-one. $23550$ stations total.
That's a lot of stations, but considering you're talking about exploring the galaxy, it's not really that bad. Plus, you only need to blanket the parts of the galaxy you intend to explore.
Modifying the number based on different view distances.
Ok, so we can already see more than 1000 light years using parallax methods. What if you want to calculate a different value?
There are three cases. In the case where $a<433ly$ (or you're looking at a spherical galaxy or something), stations is proportional to $a^3$. So take the ratio of new $a$ to calculated $a=1000ly$ above, cube the ratio, then divide that into 39000. For example, using $a=100ly$:
$\frac{39000\text{ stations}}{(\frac{100ly}{1000ly})^3}$ $=\frac{39000\text{ stations}}{\frac{1}{1000}}$ $=39\text{ million stations}$
In the other two cases, where $433ly<a<866ly$ and $a>866ly$, station count is proportional to $a^2$. Same thing, but square the difference.
$\frac{23550\text{ stations}}{(\frac{5000ly}{1000ly})^2}$ $=\frac{23550\text{ stations}}{25}$ $=942\text{ stations}$
That's the entire galaxy with only 942 mapping stations.
How do we update that in "real-time"?
Use the "sneaker net", combined with FTL, like in this question.
Basically, each mapping station has a few small FTL drones that warp back and forth between nearby nodes. A drone's host node gives the drone all its current information via some kind of close-range, wireless (or wired, doesn't really matter) transmission. The drone pops to each "connected" node (the six nearby nodes: up/down, left/right, front/back) and transmits that information to the connected node with the same kind of short-range transmission. At the same time, it would receive information about distance nodes from the connected node. Then the drone pops back to the host node and uploads all the newest data.
You'd need a pretty big dataset to hold all this information, but a society this advanced shouldn't have much trouble with that. Each node has a complete copy of the data at all times. Far away nodes will be slightly out of date, but the FTL nature of the network means they'll be accurate within about $JumpTime\cdot NumberOfNodes$. This is a linear node count; worst case scenario is going to be about $JumpTime\cdot\frac{2\cdot GalaxyDiameter}{NodeDistance}$ (data is coming diagonally across the grid, so it has to jump south, east, south, east, etc., for example, meaning the delay is about twice as long as going straight along the nodes).
For a 2 minute jump time (time to jump, transfer data, jump back, transfer new data) and 5774 ly node distance (corresponding to a 5000 ly view distance), that's $2min\cdot\frac{200000ly}{5774ly}$ $\approx69min$, or about 1 hour. The number is linear, so if you double the jump time, you'll double the lag time.
Also, it depends on how many drones there are per station. The 2 hour number assumes six active drones per station; if you only have one drone popping to 6 stations, that's six times the delay, or about 12 hours. For the 2D grid of cubes, you only need 5 drones per station, and for the 2D square grid, you only need 4 drones per station.
In a highly 3D grid (in a spherical galaxy or similar), worst case is $\frac{3}{2}$ worse, because it's doing south, east, down, south, east, down, etc., for example. Note that a disc-shaped galaxy (like ours) will still use the above numbers, even if you have a lot of nodes. In this case, the number of down jumps will be small compared to the number of south and east jumps, so you can ignore them.
What is "real-time"?
As kingledion notes in a comment, "real-time" isn't really real-time. Each node still has a pretty big lag between the events happening and the light hitting the node. My assumption was that we just needed to see the stars closely enough that we could keep the calculations accurate.
If my assumption is correct (the question seems to indicate that's all we need), then the "real-time" aspect isn't really important. You just need to update once every few decades or centuries to keep your simulations reasonably accurate. This is a good thing, because it means you can get by with far fewer drones and energy requirements.
However, if you want more accurate data, you'd need to either:
A) Have a lot of probes. To keep everything within 1 year accuracy, you'd need to bring the probe distance to 1 light year. That's around $4\cdot10^{13}$, or $40\text{ trillion}$ stations. A lot more than you likely want.
To be fair, there are only 200-400 billion stars in the galaxy, and there's not a really good reason to use more than one station per star except highly traveled FTL routes. So 200-400 billion is a more "reasonable" cap.
B) Have probes that do a lot of hopping around. Depending on your FTL energy requirements, this might be quite difficult. But you can just have (relatively) a few probes that pop from star to star. From this site about cloaking (and how you can't do it in space), they calculate about 4 hours to scan the entire sky for things the size of spaceships.
Our futuristic space probes could likely do it in 30 minutes or less, though they'll need to do three or four scans from different positions, so let's call it 2 hours (probably a lot less). You say these guys can jump hundreds of light years at a time, so a probe should easily be able to hop the five to ten light years between nearby stars in a single jump.
Add the jump time to the scan time (the probes can absorb most of this by running calculations and charging the jump drives while scanning). Let's say jump time is pretty minimal, so the total is 2.5 hours per system.
Now, let's say you want data less than 1 year old. Each probe can jump through $\frac{8760 \frac{h}{yr}}{2.5 \frac{h}{\text{system}}}=3504\text{ systems}$ per year. This means you need around a hundred million probes to cover the galaxy.
Both of these options also have the advantage that you can see all the planets and so forth inside every system. At high cost, of course.
What about those pesky, unexplored systems?
If you're jumping into uncharted territory, you'll want to send a lead drone. The drone pops in, maps the nearby area, then pops back to the main ship with the results. You just want to make sure you don't hit anything, so the drone can be tiny. If the drone doesn't come back, don't teleport there. Send a second drone a few million miles away and try again.
Space is huge. Even stuff inside the solar system is really far apart. The reality is that you could randomly jump around our solar system for the rest of your life and probably die of old age (or mutiny)1. A couple lead drones should be more than sufficient for most crews.
1Derivation of safety statistic.
The Sun is about 99.8% of the solar system's mass. The Sun's density is $1410 \frac{kg}{m^3}$. Ice comets have a density of $0.6 \frac{g}{cm^3}$ $=600\frac{kg}{m^3}$. The Sun's volume is about $1.4\cdot10^{27}m^3$. Double that and you've got way more than the volume of "stuff" in the solar system. The solar system (just counting out to Pluto) is about 7.5 billion km in radius. That's a volume of about $1.8\cdot10^{30}km^3$.
That means about $\frac{1}{1.3\cdot10^{12}}$ of the solar system is stuff. If you make one jump an hour for 60 years, that's 525600 hours. Your probability per jump of hitting something is $P=\frac{1}{1.3\cdot10^{12}}$. The probability of hitting something after N jumps is $p(n)=1-(1-P)^N$. $p(525600)$ $=1-(1-\frac{1}{1.3\cdot10^{12}})^{525600}$ $\approx 4\cdot10^{-7}$. That means about 1 out of 2.5 million people will hit something if they all jump once an hour for 60 years.
Some guys from Reddit come up with something similar.
