That's about US$130/day, it's definitely possible, but you'll have to keep a sharp eye on your budget and stick to it. If you haven't booked flights yet compare the pricing for Haneda flying into versus Narita. Assuming you're flying into Tokyo first. If the pricing is within $50 choose Haneda. It's actually in Tokyo and will be faster and cheaper to get to your lodging. Also look into flying into Tokyo and flying out of Kansai so that you can head north out of Tokyo and then make your way south and not have to return all of the way to Tokyo to catch your return flight. Again price dependent.
Youth hostels, ryokan, and capsule hotels in Tokyo and Kyoto are surprisingly inexpensive. Budget US$10-$30/night. Once outside of Tokyo the prices will be slightly higher. For example, Nikko and Hiroshima will be about US$20-$40/night.
You'll be spending as much on daily transport (subway, local trains) as you will be on lodging. Depending upon on how many sights you'll be visiting daily look into buying daily rail passes. They are typically about the same price as two short round trips within Tokyo proper, but there are different subway lines (JR, Toei) and railway lines and the daily pass doesn't cover them all.
Longer distance rail travel can cost between US$15-$300 depending upon distance and service (limited, express). Compare to flights, because it might be cheaper to fly to Kansai (Osaka, Nara, Kyoto, Hiroshima, etc) than the train. Factor transport to/from airports as well as potential luggage fees. If you're staying for a month though you should definitely purchase the 21 day rail pass BEFORE YOU LEAVE and then activate on the day of departure from Tokyo. It's Y59,350 and allows you to hop aboard nearly all of the JR trains. If you don't have trouble sleeping on trains you can also book long distance overnight trains to save on lodging.
Likewise on admission fees to attractions. Budget about US$25/day for fees.
Food in Japan can be really inexpensive or blindingly expensive. Nearly all restaurants have set lunch specials that are really filling for about US$10. Note that almost all places will provide more steamed rice if asked. A basic bowl of ramen can be as cheap as US$1.50/bowl. The convenience stores all have fresh food and you can pick up a stick of yakitori and a maki roll for US$3-4.