I would really like to go to the Kaiyukan aquarium in Osaka. It is one of the largest public aquariums in the world and holds, in the largest tank, a manta ray and a whale shark. There are 27 tanks in 16 main exhibits. wiki link
At certain times of the year the Kaburenjo Theatre in Kyoto holds performances of Geisha girls that the public can view where tickets aren't overly expensive. They showcase the skills of a geisha and this is a rare opportunity as the only times that you would otherwise see them would be at an expensive, private show. Kaburenjo theatre geisha link
The zen garden of Kyoto is also of interest to me and it is a holy and sacred place to meditate although it is a popular tourist destination so will almost never be completely quiet. Inside Kyoto zen gardens
Nijo castle consists of two concentric circles ( equally spaced circles), Ninumaru palace, the ruins of Honmaru palace, supporting buildings and several gardens. It is one of the seventeen historic monuments situated in ancient Kyoto. In 1601 the construction of it began and in 1625-26 it was completed. Kyoto travel Nijo castle link
Any shrine or temple in Japan would be good enough for me as long as it was traditional. I can't explain why I want to go to one I just have an urge to and I think the designs and intricacy are beautiful. Wiki link
I would also like to go to the Studio Ghibli museum in Tokyo as I am a big fan of Hayao Miyazaki's work and films. It is what it says it is: a museum celebrating the animations and films by Studio Ghibli.
On my visit to Japan I'd also like to go to one of the villages were you can see fireflies each year that come up to you and sometimes land on you. This would be a truly magical experience, especially at night when all you can see is the bright green glows of the fireflies. BBC Two firefly clip
There should be lots more on my list but these are just the main things that I would do. If you have any other things to do or places to see that you can recommend for me I would love to hear about them.