Ginza, for one, and the Royal Palace grounds aren't that much farther (assuming you like to walk.) Tsukiji is an early morning trip, and a tempura meal at the venerable Ginza Tenkuni restaurant is worthwhile, IMO. (Look a couple of pages back to my report for some information and links.) But don't expect a "real neighborhood" - Shiodome was not that long ago the railroad freight yards, and it's just starting life as a neighborhood, and there are nearby places like Hamarikyu Gardens that are nice walks.

You can catch light rail systems adjacent to the building or in the nearby Shimbashi station.