You may have been thinking that this trophy/achievement may have been referring to the "Special Arena" optional quests... but sadly, that is not the case. Instead, travel to the Gathering Hub in Astera, and talk to the Arena Lass (the clerk on the left) where you'll be able to pick from two options - Arena and Challenges. All the quests under the "Arena" category fulfill the requirements of this trophy, and they are essentially time-attack challenges wherein you'll be given a choice of one of five loadouts you can use to defeat the target monster as quickly as possible. You do not get to bring your preferred armor, weapons or items to the fight, so be prepared for the fight to be on more or less even terms, no matter how trivial the monster would otherwise be. Complete any fifty arena challenges - earning any completion rank in the process, and repeating any of them as many times as you wish - to obtain this trophy/achievement. One of the least-rewarding, grindiest trophies/achievements in the game, honestly.
Nowhere to Go but Up
Complete 50 arena quests.