My Tokyo visits have been equally short and this is a lot to do in one day. Skip Tokyo Tower and the Rainbow Bridge, but do go to the Metropolitan Building if it's a clear day.

From there it's a pleasant walk through residential streets to the northwest entrance to Yoyogi Park and Meiji Shrine. From the southern end of Yoyogi park it's not a long walk to Shibuya. If you're up for more at that point, you can hop on the Yamanote line and head elsewhere.

From my experience, picking a few things of interest and taking the time to see them is more rewarding than trying to run all over a huge city only to get relatively quick glimpses of a lot of things. Whatever you decide to do, have fun!