Your best bet here is to advance the game enough to get into the later tournaments where some of the arena dangers show up. Then you can go back to the earlier arenas with a new robot so everyone has lower hitpoints. Ideally pick the cat robot with the built in camouflage but you can avoid enemies as long as you stay somewhere secluded. Then you just hide and skip your turn until your opponents die from the dangers of the board. The computer isn't great at avoiding dangers, but the gas cloud and meteors are your best shot for quick kills.
Because some of this is random, it's nice to have some cash available, or enhancements you can sell and access to a store, just in case the random dangers hit you too much.