I've beat Skagzilla on playthrough 1 and 2 as a hunter, and once as a soldier. Yes, he's got a lot of hitpoints, but he's actually a lot easier than lots of the other arena-type fights.
He's reasonably easy to stay away from, except when he charges, and there's no other enemies to keep you from running like hell until your shield recharches, so just stay alive as your first priority, be patient, and then chip away at him the best you can.
Here's some points of strategy:
- Circle strafing is key. Keep moving away from him and to the left or right while facing him and shooting at him. If you always are moving to the left and away, you'll rarely be in front of his mouth when he roars and can get a couple criticals that way.
- Acid is best, fire is second best, most important is just high damage numbers.
- It's very possible to run out of ammo on your highest damage weapon, so have a couple you can switch between, at least. High fire rate sniper rifles are great, so are machine guns... both allow you to stay away as you wear him down.
- Don't wait for mouth shots. Try and avoid his shoulders and go for his legs and back, you'll do a lot more damage that way.