Reddit mentions of Salomon Men's Speedcross 3 Trail Running Shoe,Deep Blue/Union Blue/Black,10.5 M US

Run as much as possible. As others have mentioned, running hills/trails would be the most practical way to go about it. It's usually advised that you can run about 6-8miles comfortably before Tough Mudder.

Strength is certainly required, but you don't need to be a bodybuilder. Focus on doing some pull-ups and dips in the weight room and you'll be fine. The lighter you weigh, the easier most of the obstacles will be.

Make sure you get your rest. It's equally as important as the exercising. The same goes for your diet.

If you're still looking for a pair of kicks then I HIGHLY recommend these...

http://www.amazon.com/Salomon-Speedcross-Trail-Running-Union/dp/B008JEW692/ref=sr_1_1?ie=UTF8&qid=1375233013&sr=8-1&keywords=salomon+speedcross+3

Those are the best trail running shoes I've owned and they're still in great shape after Tough Mudder. The rubber cleats get surprisingly great traction on smooth surfaces. Also, I experienced absolutely no foot/ankle pain after the run.

For clothing, wear a light shirt and socks that are at least mostly polyester or whatever else wicks moisture. They will get soaked, so the lighter the material the better. I don't wear compression shorts, but I've got nothing against them. You certainly don't want anything too baggy that will get snagged on something.

The week leading up to Tough Mudder should be a week of very light training. Don't be trying anything that could possibly get you hurt. A lot of people get cramps during Tough Mudder, so make sure you're getting plenty of salt and water in your diet a few days before the run. And don't forget those carbs!

Good luck!