I paid almost nothing out of my own pocket for a 1-year exchange to Japan in college.
The school I was going to at the time had exchange deals set up with a few different schools in Japan, most of which have allocated funds from JASSO scholarships; they pay your airfare and give you 80,000 yen a month cash for ten months. There's also the US-Japan Bridging Foundation scholarship, they award something like $4000 one-time for students doing a full academic year in Japan (or a lower amount for one semester). It's pretty competitive, but you can get it if you keep good grades and demonstrate a real interest in studying in Japan.
I'm pretty sure there are other scholarships that you can fish around for.
If you do an exchange through a home university in the US, you generally just pay tuition to your own school, which can mostly be covered by whatever financial aid you were already going to take in the first place (though direct student aid (Stafford loans) for your first two years is limited, so if you don't qualify for income-based grants, you'll need parent loans or something else to cover tuition).
If you can scrounge up all of that, it'll get you by pretty well. ESPECIALLY if you go to a school in a smaller/cheaper city, and not Tokyo or Osaka. I went to Tokyo, myself, but a friend of mine went to Okayama, and got the same 80,000 yen per month that I got, but his dorm rent was less than 10,000 yen (the dorm was a crap-hole, but who can complain for something like 80 bucks a month, and who wants to go to Japan and spend all his/her time in the dorm room anyway?) so he had more than he needed.
I would recommend doing your undergrad in the States probably anyway, but there is also the Monbukagakusho scholarship, which has been discussed quite a bit on this forum. If you get that, they will pay for a year of language school and four years of college in Japan... and actually they have a one-year exchange scholarship, too, for students in US schools doing exchanges, but you're not allowed to take other scholarships if you accept that one, I think....
Hope that helps... welcome to the gang and I wish you a nice trip to Japan whenever ya get there!