Should you choose to fight, you've got 300 minutes to run into them again, or they'll have fully recovered. If you encounter them again and fight them, the counter is reset and you have another 300 minutes to find them again. If you bypass them, the timer keeps counting without being reset. You can keep track of idols you've encountered in the 勧誘リスト (the blue button on the left, when you get into the event's welcome screen) - remaining HP and time to recover are listed there.
Manage your stamina and pick your fights, too. If you're getting low on stamina, and you're sitting on a full meter, it may be best to fight the first idol you run into then rest. Your ATK will refill while you do. Conversely, if you've just blown all your meter, you can take that opportunity to rest up and recover ATK and stamina.