Konnichiwa!! Just spent 2 years living in Japan. Wish I was you returning.
A good place to start with online info http://www.japan-guide.com/
For train times throughout the entire company www.hyperdia.com
For the best priced hotels/hostels/capsule hotels http://travel.rakuten.co.jp/en/
Must see places: Tokyo (Tsukiji Fish Market, Harakuju, Shibuya) , Kyoto (Kiyomizudera, Nanzen ji, Heian Jingu, Golden Temple), Nara (a great day trip from Kyoto), and Hiroshima (I am biased I lived there- peace park, miyajima). Many would put Osaka in this list. it is an amazing city, but in terms of sightseeing not to much to do. A great to place to wander around though if you have time. oh and if going during the summer hiking mt fuji is a must! oh and on the way to Hiroshima stop at Himeji Castle.
With a month I would also look into going to Koya-san (overnight stay in a ryokan with monks), Fukuoka (just a great japanese city), possibly spending a week around th temples in shikoku (might need to rent a car for this), nikko, hitting up some onsens in nagano is great too. I never made it down to Kagoshima but would have loved to to hike around. If youhave time (and money), I would def try to go to Hokkaido.
In terms of places to stay- you can find some for 50 bucks. A cheap and really fun option is staying in a capsule hotel. I also use a site www.couchsurfing.com and met some amazing people traveling this way.
A way to get immersed in the culture is wwoofing. (wwoof.org I think). You pay 4000yen and can go to any of the farms that participate. You put in a few hours a day working, then the rest is free time. A great way to meet japanese people.
Getting around- japan rail system is unreal. By a Railpass for 1-3 weeks and it will take you everywhere.
To minimize costs stay at cheaper places, eat at conveniences stores once in a while (food isn't bad!), drink from convenience stores before you go into clubs, oooh and if up for hitching can be a kick.
Also look into what festivals are going on during your stay. During summer you can basically find festivals almost every wkd.
I hope this helps a bit!