There are a couple of really nice botanical gardens in Osaka that I used to visit when I was living there a few years ago. One is the Hattori Botanical Garden (link in Japanese only) in Hattori Ryokuchi Koen, and the other is the Sakuya Konohanakan in Tsurumi Ryokuchi Koen. I’ve always been kind of disappointed that I couldn’t find anything like them in Tokyo, but last weekend when I was in Shin Kiba, I stumbled across the excellent Yumenoshima Tropical Greenhouse Dome.
The second dome is domestic plants, and then the third one has plants from the Ogasawara Islands like these huge palm plants.
Finally there’s a little room full of carniverous plants.
There’s a PDF with contact details and instructions on getting to the greenhouse here: http://www.kensetsu.metro.tokyo.jp/kouen/kouenannai/park/english/yumenetu.pdf
Admission is just 250 yen for adults. If you’re the least bit interested in plants, I really recommend it.
While you’re there, you might want to also visit the Daigo Fukuryu Maru Exhibition Hall, a moving museum about a Japanese fishing boat that was accidentally exposed to radiation during an H-bomb test in the 1950s.