Hello from Japan
You can find a lot of useful information here:
http://www.japan-guide.com/
I haven't used a weekly/monthly mansion for 7 or 8 years but at that time they usually had Internet access already, so it should be no problem at all nowadays.
October/November and March/April/May are usually the best months to be in Japan. If you go for 2-3 months I would actually stay in several cities (e.g. Tokyo, Osaka, Kyoto, Sapporo, Fukuoka, etc.) and maybe in a smaller city somewhere as well.