Not too far away from Asakusa is Ryogoku. If I were to recommend one museum in Tokyo it would the Edo Tokyo Museum. You should also have the opportunity to have a quick look at the Sumo Stadium and it's small museum (free) which is close by. There is also a pleasant Japanese garden in the vicinity, the Kyu-Yasuda Teien Garden.
I was going to suggest Kaeruhonpo for evening monjayaki (unique atmosphere, inexpensive and the owner speaks excellent English) but their website address is no longer valid, so cannot be sure it is still open. They also did great 500yen deals at lunch time.
Keep your eyes open and your nostrils flared, very possible that you'll catch a glimpse of a passing sumo wrestler, walking or on a bicycle. Note the smell of their hair pomade, glorious.