So, I didn't see an arena thread, so here goes.
After some trials I think I have the training system figured out. Not how numbers or anything like that, but, I have figured out that the training depends souly on time. Unfortunately you can't just leave your game on and go to sleep, as after I think 20 mins, you can get your monster back, and it won't advance any further for any amount of time after the 20 mins. It also has nothing to do with amount of times you time travel, or amount of times you have saved. Also, I only think its 20 mins, I haven't actually timed it accurately yet.
Anyone have anything to add about the arena?