The word you are looking for isn't 'hostel' btw; it's 'dorm.'
The dorm I am living in costs 32,000 per month and does no include meals. You are allowed to live there for two years, so if you move to a graduate program, you will have to move. There are, however, other dorms available that would allow you to avoid key money/realtor fees.
I will list the monthly fees I am looking at to give you an idea.
cell phone. cheapest plan possible. about 2,500 per month.
Internet: about 4,000 yen per month
utilities, dorm: maybe 3,000 per month
transportation: for me, round trip train fare to and from my school, 20 times a month, would cost about 10,000 yen per month. I also might have to purchase a bike parking spot at the station that would cost another 2,000.
-textbooks. I don't remember for sure, but I think we will have to pay for textbooks. Averaged out per month, probably something like 6,000 yen.
-various cleaning/apartment upkeep expenses. 3,000 month.
Combining all of these things, that'll put my base expenses, which does not include food, entertainment, or going anywhere besides from my school, at around 60,000 yen per month. Once I eventually move to a private apartment, I would probably save on transportation but spend more on utilities and rent, so at that point it'll probably be more like 90,000 per month.
You will be able to travel a little around Japan, but remember that transportation/lodging is quite expensive in Japan. It would probably be hard both to do much traveling and go out and have fun. But you can always get a part time job...
ETA: You will probably not spend any money as gifts during the Japanese holidays. That is not a role that foreign exchange students are expected to fill. If anything, you might receive a tiny bit of money, but I wouldn't count on it. Omiyage will similarly not be expected since you won't be working at a company...