I think the other posters have given good suggestions already, but I'll post what I found out about adoption of a foreign national.
Under Japanese law, adoption is not going to give you Japanese citizenship. You still need to go through the naturalization process that may take (at least 5) years. It has a lot of paperwork involved, gathering documents and certificates both in the US and Japan. So, you will have to deal with visas the same way other foreign nationals do during your first 5 or more years in Japan.
People of Japanese ancestry have a special visa category (Nikkei-jin), but most who apply are from China, South America, etc. Here, a lot of paperwork is involved, since you have to prove that you are certainly of Japanese ancestry. You will need lot of copies of birth certificates of your family to trace your blood line to your Japanese grandparent. But, once you get this visa, you're able to study and work here.
Both of these are troublesome but allows you to extended stays in Japan. However, these visa statuses do not grant you preferential treatment for university entrance. Since you are not an exchange student, you may have to compete against native-speaking Japanese students on an equal footing, which is impossible if you don't have near-native fluency and exhaustive preparations for the entrance exams. There might be special programs in some universities, but you still have to pay your way through it.
If you come as an exchange student, you are treated that way, and you fall in a category apart from the Japanese students, gaining access to special Japanese language programs for foreigners to help you assimilate into the Japanese study environment. As Ewok has suggested, you might be able to gain scholarship of some kind to fund your study in Japan.
So, all of these options have their positive and negative aspects, and the details must be checked out thoroughly. (What I wrote here is very general and there might be errors in what I think I know.)
As Mike said, it is best for you to consult your nearest Japanese embassy or consulate about visa status and do some research on exchange programs and financial aid.