In terms of ancient culture I'd recommend Kyoto and Nara.But there are small temples, shrines etc. everywhere in Japan.
And of course, the modern day Japanese shrine -- the Jidouhanbaiki (自動販売機: vending machines!) \(^O^)/
Don't forget the Ghibli Museum if you are a fan of Ghibli
ps. that's in Tokyo and you do need to pre-book your ticket(s)