In order to get this achievement you will need to be in Astera. Astera is the main city with the smithy. Once there go to any one of the lifts around the map. If you're having difficulty finding them you can use the mini map (Press the Back Button/Option Button).
Take the lift to the 4th floor, called the Gathering Hub. This is the social area and there are several things to do here. This is where you can locate the Arena.
Talk to the Arena Counter in the gathering hub and accept one of the challenges. Once you complete the objective in the challenge the Achievement should pop.
The easiest way to beat this quickly is to select the first challenge, against the Pukei-Pukei, and pick the hammer class. There are several large trees with nets on them. If you Hold RT and walk up to these trees in will trigger the Hammer Air attack. This will send you character flying doing a ton of front flips. It does a lot of damage if you can hit most of the attacks and especially the last one. The Pukei-Pukei will struggle to hit you doing this attack and you should be able to mount him several times. Just keep on flipping at him until he dies.
Please comment below with any questions.