Ya $300 for a months worth of food is cutting it close, u cant eat junky stuff the whole time your there. But, if your trying to budget, 7-11's actually has decent food there (unlike the USA) You can get a decent meal in there for under $5 or 500 yen, and they'll heat it up for you too.
Must see places in Tokyo:
Asakusa Temple (not to far from Kuramae)
Theres a sumo museum/arena not to far from Asakusa temple, i don't know the name of it but you can walk there in like 30 minutes from the temple.
Tokyo Palace is open on Sundays i think
Tsukiji Fish Market Is an absolute must!! And bring some money for food. The best Donbori i ate in Japan
Another good place to go on Sundays is Yayogi Park, there's usually lots of people dressed up and stuff.
Meiji Shrine is nice also.
Tokyo Tower is cool.
Roppongi Hills Tower is awesome. Not sure how old you are but theres lots of bars and nice restaurants there. But mainly, there s a panoramic view of Tokyo city witch is stunning as well as a contemporary art show for 1500yen. IF you can get to Roppongi hills tower before 7pm you get to go to the sky deck/bar for the view... Its awesome.
You can take an hour and half bus ride to City near MT.Fugi is (Fujikawa) from Shinjuku Station for 1700 yen ( if i remember correctly).. Then from Fujikawa station another bus ride to Mt. Fuji for an additional low cost. You can do that in a day trip or make it a 1 or 2 day thing because theres more to see in that area, and the city is beautiful. You can stay at a hostel for less than 3,000 yen a night so if you have some extra cash its well worth it.
Thats all i can remember for now, and im not sure if i spelled everything correctly. Other obvious places to go are Shibuya, Akihabura and Harajuku..
Hope this helps.