This achievement is very self explanatory. You just need to stay awake for 24 in-game hours (24 real time minutes). The only stipulation is that this 24 minute timer starts at 6 AM. After that... do literally anything except for going to sleep and this achievement is yours! I would suggest you get this achievement in tandem with While You Were Away.
Here's my video for every achievement if you'd prefer. Skip to 6:01 for Up All Night.