In the near future (next few decades) a giant spaceship is detected as it decelerates on its way into the solar system. It heads for Earth orbit, but a few weeks before reaching Earth, it veers away and ends up in orbit around the sun not far from Earth, about 10 times the Earth-Moon distance.
The ship doesn't respond in any way to attempted communications. NASA re-purposes a satellite that was scheduled to launch for Mars and approaches the ship. To everyone's surprise, the ship opens a large hangar door and sends some radio signals to the satellite. After some time and repeated signals, the door closes again. Next an international team goes and boards the ship.
It turns out to be a generation ship but evidently failed, as it's full of dead human-size insectoids and only a few cryogenically frozen specimens, most also dead. The ship seems to be otherwise intact and is run by some kind of computer. Unfortunately, the computer is not compatible in anyway with Macs or any other human-made computers. With some advanced cameras, the displays can be converted to human vision and they appear to show that Earth has been rejected as a colony world and the next destination is an Earth-like planet most likely some 50 light-years away. There's also a countdown of uncertain length, but likely to be 3-4 months because that will be when the trajectory towards the next star is optimal.
Some humans decide they want to grab this once in a lifetime chance to hitch a ride on this ship, despite the risk of meeting the same fate as the original crew.
The technical part
The aliens' basic physiology is compatible with Earth-life (the very reason they headed for Earth) and the atmosphere on the ship is readily breathable by humans. The water dispensed in the ship has way too high mineral content, but nothing that would destroy human filters and some specific taps supply demineralised water.
However, the plants in the hydroponic farms are all toxic to humans and while the light seems good, the water is full of bacteria that kill any other plants. They need to be flushed and run manually somehow. The waste-recycling method is unknown (it can be adapted to make things more workable).
The ship itself is designed in a very distributed and resilient way. Attempts to remove subsystems and attach human systems to their power supply points work once it's figured out the ship supplies DC electricity at the roughly the voltage indicated by the system upon connection. Unfortunately, no control connection seems to work, nor does the ship's computer interact with the systems in any way.
The computer running the ship seems to have been programmed in a very permissive way. It doesn't take any hostile action and doesn't override or reverse non-damaging changes to the ship. It appears to recognize humans as living and treats them as if they were regular occupants, activating lights, heating and other functions on demand. It even responds to certain sounds, but the reply is always the same, most likely "I don't understand your request, try again".
However, it doesn't recognize anyone or anything as having the authority to access or change the reactor and drive section nor the navigational systems. The same goes for the shuttles and landing craft in the hangars. The ship is going to depart and short of destroying it nothing will delay that.
Humanity has one huge leg up: They have established a small base on Mars and were just preparing a serious colony mission consisting of 2 passenger ships holding 50 people each and several cargo ships to deliver machinery and supplies enough to set up a colony for those 100 people. The alien ship is much closer, so multiple trips per ship are possible.
The sponsoring governments are loathe to postpone the Mars effort, but if the chances for surviving on the alien generation ship are good enough, they are willing to divert some or all the Mars colony ships.
What resources do they need to take to the ship and what will it take to get them there?
In response to comments and answers:
The time limit is flexible in the service of storytelling. I obviously chose it too short, but it should not allow for complete preparation and a 99% certainty of survival. That said, for all intents and purposes a single person on board with enough supplies could live out their life on the ship, it seems to be that safe. But the new crew is going to have to get aboard with some stuff that they have to figure out on the way if they want to make it work in the long run.
A lot of equipment is ready to go, thanks to the Mars mission. That probably includes a whole set of Hydroponic-ready plants that cover nutritional needs (though only tested up to few years with adults). Of course, the equipment was meant for only 100 people when at least ten times that will go on the alien ship.
Of the cryogenic pods, only a few show status lights that appear to indicate all is well. The insectoids can't be revived yet (an attempt will result in the alien's death) and are not playing an active role in the story, but their presence obviously does. (Thanks liljoshu!) All the other insectoids were frozen where they died when the ship lowered the temperature to somewhat below zero degrees celcius. It's unclear what they died from, but there is no sign of disease or violence. There is also no evidence of a queen or other special types or aliens. The exact cause remains unknown during my story, though the leading theory becomes that they were hit by an intense radiation burst.
Finally, one of the landing ships is pulled from the hangar to a safe distance and experimented on to try to disable the computer or drive system. It does not end well, so the decision is made to not tamper with the much bigger versions on the generation ship.