JET is for people with degrees to come to Japan for 1-5 years (usually 3 max) and be ALTs. Key point is that you need a bachelor's degree first.
If you want to study at a Japanese college, whether for a year or more, you have to decide what major it is. A very few unis here will have classes taught only in English. Otherwise, just how do you expect to understand any lecture or textbook when you have a working knowledge of only 350 kanji. Imagine knowing how to spell that many English words and thinking one can attend college. (Just for the sake of comparison, American college students have a vocabulary around 20,000-25,000 words.) How are the classes at those unis that interest you?
By intensively, do you mean only study japanese, like at a school designated only for japanese? University credit does not count?
It works like this. You get accepted to the uni, but they require that you take intensive J language courses for a few months first. Intensive means 6-8 hours a day for 5 days a week during those months. The intention is to drill spelling, grammar, and basic conversation into you. I know you are already studying it, but you may have to do this anyway. It's probably better for you than a class or 2 per week, which is probably what you're getting now. The down side is that a few months may still provide you with insufficient communication skills to survive a classroom situation. Foreign students where I work tell me that is the case for them. Fortunately for them, they are almost all grad students and can get by separately with their own little "gaijin ghettoes" at the school, and teachers will just give them a handout or web page in English to take the place of what the Japanese students are expected to learn in class. I guess that's unfortunate, too, if you want to actually LEARN something.
To get a student visa, you have to prove you can support yourself. Keio says this, as an example:
"Be sure you can financially support yourself during the period of study at Keio. The average cost of living expense is at least JPY 130,000/month if you live in university-subsidized student housing. "
EDIT:
From Keio U site:
Japanese Language Requirement
There is no Japanese proficiency required to apply to be an exchange students at Keio University. Students are required to take a Japanese language placement test run by the Center for Japanese Studies at the beginning of each semester in order for them to register for the appropriate learning level for Japanese language courses
Applicaton Guide for Keio Exchange Program|Keio University International Center
From Todai site:
To enroll in a faculty of The University of Tokyo (four years for graduation, and six years for graduation in the school of Medicine in the Faculty of Medicine, the Veterinary Medical Sciences Course in the Faculty of Agriculture and the Department of Pharmaceutical Sciences in the Faculty of Pharmaceutical Sciences), you are required to pass the same general screening test as Japanese students or a special screening test (for students who have graduated from foreign schools). This document explains about the special screening test. Most of the lectures in the faculties of The University of Tokyo are delivered in Japanese even for foreign students. Taking this point fully into account, foreign students must master the Japanese language before they enroll in The University of Tokyo.
東京大学 | 国際化教育支援室
From Kyushu site: You have to pass a reading test.
Kyushu University International Affairs Department | Admissions