Really? Right in the middle of the worst of the summer heat/humidity AND during the height of Obon? Sounds like a potential horror show, especially for travel in and around the country.

Can you change your dates?

As for favorites, I'd say Hibiya Park near Ginza for a nice place to sit and enjoy for a bit, as well as Ueno Park for site-seeing. The multi-building National Museum complex is in Ueno Park too, right across from Ueno Station. I'd also be sure not to miss Hama-rikyu Imperial Garden, near Tsukiji.

My favorite museum is the Edo-Tokyo Museum in Ryogoku (Sumo Town), which is one stop past Akihabara on the train. While there, you could stop in at the tiny sumo museum right next-door at the Kokugikan Arena if you wish.

There are temples all over Tokyo. One of the classics is Senso-ji (Asakusa Canon) in Asakusa. There is a nice water-taxi that travels between Asakusa and Hama-rikyu if you decide to see both.