We were in Tokyo & Kyoto a couple years ago about the same time – very hot & steamy, so be prepared for that. Something that we messed up on – didn’t get tickets early enough – was the Studio Ghibli Museum. Accessible by public transportation (we did not try to drive anywhere; we did one of the rail passes) but not in central Tokyo by any means. http://www.ghibli-museum.jp/en/
The place I enjoyed most near Tokyo was Nikko. http://www.nikko-japan.org/ About an subway/hour train ride. Lovely, lovely, lovely.