You could check out a baseball game, there are numerous temples to visit, numerous museums, tons of shopping, and and abundance of excellent restaurants. Things I enjoyed in Tokyo are Tokyo Tower, Meiji Shrine, Sensoji Temple, Kamakura Buddha, Yakitori under the tracks by Yurakucho train station, Tokyo Giants baseball game. The Tokyo LDS Temple is easy to get to though not really that unique among temples but worth stopping by to get a photo. Arisugawa Park across the street is nice. Chinatown in Yokohama has the best Chinese food I have ever had.

June is likely to be somewhat rainy so plan accordingly. I would look into bringing a light rain jacket that you can fold up and put into a small day pack or something. It won't rain all day but when I worked there we had a shower or 2 every day.

I think it is well worth it to pick up a copy of the Lonely Planet guidebook for Tokyo. Also download the Tokyo Subway app as it is an excellent resource.