Master Gee has an almost impenetrable shield 4x Conference Call + Bee shields won't make a dent. In fact the Bee is a very bad idea as it's almost impossible to not be taking damage for most of this fight.
Baby Maker reloads aren't much use either. The only thing that seems to hurt Gee's shield in any useful way is his own worms.
Master Gee spawns sand worms - one at a time at first, but more and more as the fight continues.
Each worm that you kill produces a fountain of corrosion on the spot where it died. If you do nothing about this a pool of corrosion will spread out from it until it fills the entire arena and you'll be taking constant corrosion damage.
The only way to get rid of the fountain of corrosion is for it to be absorbed, either by a player (normally an instant FFYL) or by Master Gee (the only way to kill his shields).
So this fight is always about kiting, luring, singularity/phase-lock pulling, or otherwise getting Gee to run through the middle of the fountain left by the last sand worm killed before too many others spawn.
If you end up with two fountains too far apart then your fight is usually over - one or the other is going to spread until the constant DPS kills everyone.
In a word: annoying.
Is there a quick way to kill him? No, I'm pretty sure he's been designed to be very tough to defeat no matter how levelled/kitted up the team taking him on is.
That said I've seen him killed in a single shot by a level 999 player who was obviously hacked, but I'd avoid hackers and public games until the graveyard exploit's fixed.
The Wiki says there are 5 new weapons introduced in the DLC. There are four new Uniques (blue): The Cobra, The Pocket Rocket, The Boom Puppy, and The Kitten, as well as a new E-Tech shotgun called the Slow Hand. There are no new Legendaries sadly. All of the orange Tourgue weapons in the vending machines can be found as possible drops in the general campaign.
Best Answer
The challenge is bugged as of now, February 2, 2013, game version 1.3.1. (Note that fixes can arrive via network, without actual update to game files.)
It only registers if Master Gee is killed by weapon damage, not acid from worm pools. As Master Gee has absolutely insane amounts of health (speculated to be in the billions), it is very hard to do in TVHM.
It is, however, quite possible to achieve. In Normal difficulty, Master Gee is level 32, and his health pool is much more manageable. It is recommended to use your best level 50 damage-dealing build with The Bee to minimize the chance of acid stealing your kill. It is still a possibility, so don't get too upset if that happens. You can put less stacks of acid on Master Gee, but this would mean more live worms -> bigger chance to get your Bee scratched at the worst moment.
After the kill is registered and the challenge advances, the arena door opens. It is then possible to leave the arena and return, which respawns Master Gee. By switching to operating system and advancing the time one day ahead, Master Gee can be fought immediately again.