The Qwarktastic Battle is unlocked after beating all the other AN challenges. Once you do so, the Qwarktastic Battle will open up. It helps to have the Qwack-O-Ray, quite a few Level 5 weaons as well as 30 minutes of time, as the battle is 100 rounds long. I suggest waiting till near the end of the game to do this for that reason.
For the first 30 battles, keep turning the enemies into Ducks with your Qwack-O-Ray to save ammo and do so quickly as there will be random time limit rounds. For the next thirty, you'll have random Sleeping Gas rounds, so continue turning enemies into ducks to save ammo.
Once you get to round 50, you'll have to fight the Twins. I suggest using the Level 5 Flux Rifle, as it can take them each down in three or four hits. If you do so quickly, you won't lose that much health.
Round 60 throws you a curveball by giving you weapon change rounds. For this reason, I suggest you have a lot of Level 5 weapons, since you can't choose which weapon you want. Keep an eye on your ammo and once it goes back up to normal, switch back to your Qwack-O-Ray to save your ammo (weapon change ammo is free).
Once you get to round 100, you'll have to face Scorpio. I suggest using the upgraded Annihilator, as the impact and numerous mines will take his health down in seconds (and your Flux Rifle is probably near depleted from the Twins fight). Once you destroy Scorpio, you'll unlock this trophy as well as 200K bolts (2 Mil in Challenge Mode) and a floor trophy for the Starship Phoenix. While this one isn't as tough as the Impossible Challenge in Going Commando, it's still worth patting yourself on the back for.