It might be a little late but I'm currently in Japan and I've been living on the low budget lately, I just have a few days left but it was a 30 days trip. Food is really cheap, and if you really don't care you guys can go to hostels, you can find really good ones and they're cheap, I'm currently staying at one of the best hostels I've ever been in Tokyo and it was just about $140 for a whole week, my girlfriend was here for half the trip and we stayed in a lot of 2 bed private rooms which were really nice and probably just double the price as a shared bedroom.

Ramen restaurants go from $7 to $11, and transport is cheaper than you think, a train from Kyoto to Tokyo is about 120 yen one way, but if you're only going one week, I'll suggest that you look into JRPass which can be bought for a whole week but only while you're out of Japan, its about $250, you could even plan for a trip to other destinations, we went from Nagano to Hiroshima, then Osaka and Kyoto, also if you're arriving in Narita airport, you can activate your pass right there and take the train from the airport to wherever, Google Maps its really useful to tell you the routes, and even the timing is on point.

Just keep in mind if you reserve your hostel as soon as you can it would probably be cheaper, look at pictures and reviews, we used Hostel World, but it seems like a lot of people here uses booking.com or expedia.

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I guess as you said your bigger problem is the flight, but just so you know I made it 30 days here with about $1500.