It's a tall order...
Based on OP's description of the AI, it has no prior knowledge of space travel, engineering or how to sustain life. While I believe that this knowledge could be gained over time, it would require extensive human assistance to interpret and filter the available data. The crux of the issue is that the AI needs feedback on whether it did a task well or not.
Thus, its ability to acquire new skills is limited by how quickly humans can guide it through the process, and by the teachers' ability to tell fact from fiction. Training the AI to do anything useful is going to take far too long.
Now, since, in this apocalyptic scenario, humanity's not going to complain, there is the option of hijacking existing infrastructure (aircraft, satellites, factories etc.), which would give the AI an opportunity to gather practice data on its own. For simple tasks, like flying a drone from A to B without crashing it, minimal supervision would be required. Assuming your AI's a master of abstraction, and careful preparation and suitable know-how on the humans' part, it might be possible to automate some of the tasks to, say, execute a rocket launch with only a skeleton crew. You'll need some humans in any case because
- Suitable robotic chassis for every step of the process don't exist.
- There will be gaps in the AI's knowledge you didn't think of.
- You don't get any trial runs. One ready-to-launch manned space mission just waiting for takeoff is already incredible luck.
...and getting there was the easy part
So, congratulations, you got incredibly lucky and managed to launch a handful of people off the planet. Now what? You need to keep them supplied (no matter where they're going), so you have a few months, maximum, to teach the AI how to run an entire planet, from farming and mining to building space probes, on its own. Oh, but first have it mass-produce a humanoid, battery-powered, wifi-enabled chassis that can push all the buttons, move the crates and drive the tractors. You did remember to prepare that, right?