Here is my 100% Guide for Deliver Us The Moon!
This is towards the end of chapter 1.
After you turn the ignition for the second time (First time doesn't work, you have to go and do a small puzzle to get it to work), you then have 1:30 to get to the rocket to take off.
It is plenty of time, so follow the only linear path, go up the lift, and as you push another button you see a set of doors open at the end with a ladder in it.
DO NOT go up that ladder until there are about 4 or 3 seconds left on the clock.
If you go up there with 5 or more seconds remaining, achievement will not unlock.
Go to 15:15 in the video to see how it is done.
Hope This Helps!