The Imperial Palace grounds are a nice walk, and you can see the Edo period fortifications and moat, but you can't go inside any actual buildings except the tiny art museum that shows a rotating collection of items from the emperor's collection and perhaps the small theater if there's something going on.

The Sensoji area is the closest you will find to "old Japan" in Tokyo, I think.

If you go to Tsukiji, make a side detour to the Hama Rikyu Gardens and from there, catch a boat up the Sumida River to Asakusa. It's not scenic by any stretch of the imagination, but you see a new view of the city, with its jumble of old and ultramodern, and if you look to the east bank of the river, you will see the remnants of the old canal system that laced the entire city in the old days.