Right now Kingdom Hearts Unchained X allows you to pay to unlock Munny Medal Quests, but they also unlock for free in 30 minute blocks at various times each day.
As best as I can tell, the unlock times are 9:00 am PST, 1:00 pm PST and 9:00 PST (please leave a comment if your experience is different).
If the app is open at that time, that’s when your 30 minute countdown will start, even if you’re not actively playing. HOWEVER, if the game is not open, the quest countdown will start the next time you open the app. Therefore, don’t worry if it’s 10:00 am and you think you’ve missed your time slot. As long as you’re opening the app for the first time within that unlock period, you’ll get your full 30 minutes. Be aware though that it seems the option does go away after an hour or two, at least for the one at night.
Bonus Trick: If you don’t have enough API to complete an unlocked quest, simply select that quest but don’t click Begin. Press home to return to your main phone screen (no need to leave the app open) and reopen the app once enough time has passed for your API to have restored itself. Even though there will be 0 minutes left in the unlocked quest, you’ll still be able to do one more quest as long as you don’t leave that questing screen.