I would recommend staying away from Westport. There are some decent bars and restaurants, but violence and shootings there are not uncommon. River Market is a very cool area, and Zona Rosa (up I-29) and The Legends (west on 70) are neat shopping, dining, and drinking areas about 20 minutes from the arena. The Power and Light district around the arena though is the best area in the city for food drink and fun, and everything is within walking distance. Ameristar Casino is also very nice, about 15 min. north. Also, all streetcars and buses in the metro are free.
If you happen to get lost, don't stray too far east of the Power and Light-Midtown-Plaza strip. Some of those neighborhoods can get real sketchy real fast.