Arena battles can be completed by either killing all enemies or killing all enemies within 40 seconds.
There are 4 secret arenas on the first level.
Halfway through the first level there will be a well. When you enter, you will press to push a rock. This rock will turn to point one of 3 doors, and, in turn, will unlock that door in the real word. The door nearest the well (the first 1 you unlock) will have an arena in it.
The middle of the 3 doors will have an arena in it.
In the next room, there are some steps on the left. Walk up them and press to open the door on the opposite side of the room. Inside this door is another arena.
Few rooms along when you enter there will be stairs down to your left and right in between these stairs (below where you enter) is a well where yet again you can unlock a door in the real word which is a little futher along from the well. Go through this door, and follow the long path to a lift that will take you to the 4th arena.