Well, that should be entertaining! The Chicago Urbanathon is a nearly 12-mile racecourse that winds through Chicago. Nothing unusual there. What sets the event apart is that spaced along the route are a series of obstacle courses. Participants hop through tire barricades, walk along the tops of Jersey barriers placed end to end, crawl over cars, and more. See: It's urban, it's an -athlon. Urbanathlon!
As for shoes, I'd probably go with either a cross-trainer or a trail-runner. Something with a little more support than a pure running shoe. Besides, this is an event that's as much fun as it is about time, so a little extra weight won't hurt. Maybe wear something cool, like the Reebok ZigPulse cross shoe ($100), which has a wild wavy sole made of orange foam. But it's actually a practical shoe as well, with lots of sole flexibility for all the strange maneuvers you'll be making, plus good grip and support.
In a trail-runner, go for something like the Merrell Refuge Core ($110), which has leather and suede uppers for durability, a nylon shank for support, and an EVA footbed for comfort. Plus they have a nicely urban look. Or get a pair of Puma Complete Nightfox TRs ($95). They have a rock shield plate for foot protection, grippy outsoles, and breathable but water-repellent uppers. Plus they're cool-looking: black with red and yellow trim. I also like the Tecnica Wasp Low ($99), which is a little lighter shoe than the other two, but still have the support and toughness of a good trail-runner.
Time to start training!