This site allows you to lookup the possible opponents you may face in FE3's arena.
Fight selection is random, but is fixed after the fightmonger shows you the bounty. By combining some knowlegde about how the RNG selection and subsequent calculations work, it is possible to infer what opponent you may end up fighing. Occasionally it's possible to 100% predict the enemy class, but sometimes it's only possible to narrow it down to 2 possible opponents.
For more details on how enemy selection and bounty calculation works, see this page.
The subpages under Class Matchups display a list of possible matchups for each playable class that can enter the arena. Clerics and Dragon classes cannot enter the arena, so they do not have oponent data. Some classes share an opponent table with other classes. Classes which share tables are combined into a single page.
Each page has a "Player Level" and "Bounty" input that allows filtering the list. "Player Level" should be the level of the unit entering the arena. "Bounty" is the reward advertised by the fightmonger.