Another cultural recommendation is bunraku theater. The puppets are amazing, as are the narrators, even if you can't understand what is going on.

If you are going to be anywhere near Nara in October, I would recommend the Shosoin exhibit, which displays 1200-year-old artifacts from Japan and countries along the Silk Road, all kept in a 1200-year-old storehouse that somehow manages to have perfect interior conditions for preserving things, with a selection removed only once a year for the public to see at the Nara National Museum.