I would like to experience staying at a ryokan, Japanese traditional inns. You can experience to be in tatami mat room and sleep in a futon. Although ryokan in Tokyo are quite few , there are few options in Asakusa and Ueno.
Some of the places are listed here:
I think Shinjuku is the best place to stay in Tokyo because the Shinjuku station is the central for wherever you want to go. There also many restaurants and covenience stores so you will never be inconvenience. The hotels are rather expensive in that area but there're reasonable ones if you resarch.