Since you'll miss the Edo-Tokyo Museum, I'd recommend either the Shi-tamachi Museum or the Fukagawa Edo Museum. Both focus on Edo culture of the common people and have hands-on components. They also have special exhibitions focusing on aspects of Japanese culture. [Note that there isn't a hyphen in the name of the museum that starts with an S, but the filters won't allow the post to go through with that string of four letters. Thanks to our esteemed webmaster for helping me figure that out.]
The Tobacco and Salt Museum might be of interest for the quirky window that provides to Japanese culture. The Sumida Hokusai Museum has works of art in their special exhibitions and more of a cultural/historical presentation in their permanent exhibition.
All four of the above museums are in Tokyo.
If you want to learn more about the late 19th and 20th century arrival of Westerners, the Yamate area in Yokohama would be interesting.