It's not as close as the places on Randolph Street, but for what it's worth my buddies and I usually go to Club Lucky (1824 W Wabansia) before a Hawks or Bulls game, and then hop on Damen Ave straight down to the United Center.
Try Westend. It's around Racine and Madison down the street from the UC. It's bar food done pretty well with some nice seafood selections on their special card from time to time. they also have a free shuttle to the UC for events. Call ahead to make sure.