There is no time limit. A few general tips:
1) Balance private and public favours as much as possible so you don't accidentally lock yourself to one ending path
2) If you upset a girl when asking for a favour AND the heart turns black, she will need to be upgraded to the next tier to proceed. If the heart is not turned black you can ask again after her mood resets, but:
3) Do not upgrade the favour level until you've seen every option, as a higher tier will change lower level interactions as well.
4) When you get the warning about public favours near the ending SAVE so you can go back, if wanted.
5) After getting the private ending you can choose to keep playing and do all the public ones, if you want. Or start a New Game Plus.
For Luna, decide if you want sub (Looney) or dom (Bisexual) Luna.