The action always happens in an arena, but arena can also be a physical building
For your specific question, the correct term is always:
They fought in the arena.
This is because the arena is whatever area the fight happens to be taking place within. Whether that's a circle drawn on the floor, or a full fighting ring.
However, it's worth noting that if you mention a specific part of the area - you could use on, as you are referring to that location and not the arena as a whole:
There was blood on the arena floor.
This said, an Arena (2.b) can be:
a building containing an arena
As such, if you are speaking about the building (also called an "arena"), you could talk about people on top of it, or anything else happening to it.
As a side note, although the actual action happens within an arena. Things may happen external/around it.
The audience's gaze fell upon the arena.
The lights dimmed on the arena floor.
The crowd gathered around the arena.
The announcer filled the arena with sound.