Can I set Audacity to record at a certain time?

Yes! See Timer Record in the Transport menu.
You can also make Audacity stop recording after a certain time limit without using the Timer Record feature, by following these steps:

1. Turn on “Play existing tracks while recording” in the “Audio I/O” section of the preferences.
2. Choose “New Audio Track” from the Project menu.
3. Zoom out if necessary, then click and drag to select the amount of time you want to record.
4. Start recording. Audacity will stop recording automatically when it reaches the end of the selected area.