I concur. Even more impressive that it's totally private--and when we went the main guy who found and excavated it was there to answer questions at the end of the tour. Steamboat Arabia Museum
Some pretty good art museums if you're into that sort of thing--Nelson-Atkins is the big one, or there's a Kemper one that I'm pretty sure is free.
Can't help much with food around downtown. Was fairly impressed with Jerusalem Cafe and Blanc blanc burgers + bottles when down there visiting friends. Had some good Indian food in Westport, too, but don't remember what the place was called.
I'll cast another must do for Steamboat Arabia.