You can't tour the imperial palace, but you can walk through the grounds via Hibiya Park and visit the little museum that contains a rotating collection of the imperial art collection.

In Ueno, the Shitamachi Museum offers glimpses of life in prewar Tokyo, complete with rooms and "shops" furnished with authentic period artifacts.

The town of Kawagoe offers a neighborhood of Edo-period buildings that survived somehow.

But why not buy a guidebook such as the Insight Guide to Tokyo and discover things that we have never even heard of?