This is what I feel, based on the places I've been to.
my favourite free spots:
- Fushimi Inari Shrine, Kyoto
- Ramen museum, Ikeda in Osaka (visiting museum is free, but need to top up 300yen if making own cup noodle)
- Odori Park, Sapporo in Hokkaido (esp during Snow Fest in Feb)
- Asukayama Park, near Oji station in Tokyo (esp during sakura season)
- Ueno Park, Tokyo (esp during sakura season)
- Kawasaki Daishi Temple, Kanagawa
- Sensoji Temple, Asakusa in Tokyo
- Hiroshima Peace Memorial Park, Hiroshima (Museum will need to pay though)
- Osaka castle, Osaka (free to enter castle grounds, need to pay to enter castle)
- Okayama castle, Okayama (free to enter castle grounds, need to pay to enter castle)
- Bamboo grove, Arashiyama in Kyoto