Minimal? Depends on what you mean. I've both travelled in Japan extensively in a genuinely minimal fashion, but I can easily occupy myself with things that don't cost much money if needs be. It may not be for you.
This is based on what I would do if I were to visit one big city with a few out of town day trips (not too far e.g. Tokyo to Kamakura or Nikko, Osaka to Himeji). The only way to be truly economical with accommodation costs is to limit yourself to one city. Understand if you want to travel around it's going to cost A LOT more.
1 month in a gaijin house (monthly mansion) - 80,000 to 100,000 yen for a self-contained apartment, much less (50,000?) if you're willing to share kitchen, bathroom, etc. with others. Companies like Oakhouse, Sakura House, etc. are decent enough. It's more expensive than renting a normal apartment like you would if you lived in Japan, but still only a third (potentially less) the price of staying in a business hotel for a month)
Food - 60,000 yen (2,000 per day). Most people who visit and stay in a hotel are forced to eat out almost all the time. You're staying in a gaijin house which will have kitchen facilities. If you can cook then you can feed yourself for less than 1,000 yen per day, but you're travelling and probably want to eat out sometimes so I've doubled it.
Transport - 30,000 yen (1,000 per day). Again, you could get away with less and it totally depends on what you want to do. I could easily enjoy myself for a month in a big city with less than 1,000 yen per day for transport.
Everything else - Basically impossible to say but anything from 30,000 to 140,000 yen depending on what you want to do, spend money on, etc. In my opinion 5,000 yen per day for 30 days would amount to a really good, fulfilling trip.
Adding all this up it could be anything from 170,000 to 330,000 yen not including flights. Even at the higher end I would consider this very reasonable for a month in a country like Japan. I can (and have) done it for well within this range, and always have the time of my life, so even though some people will tell you Japan is the most expensive place on earth it's totally doable if you're anywhere near as easy to please as I am.
I am also currently planing my next trip (a month, like you) and my budget is 360,000, but that's because I've decided to splash out a little bit to reward myself for something I did last year. I have done long-term travel in Japan for less than 200,000 per month though.