I got some good news about this for my own case today so I'll write about it here. There's a few different ways you can enroll in Japanese studies in Japan, thus getting a student VISA and getting to live there.

1. Exchange program from your home country's school that already offers it. If you're a university student, you'll pay your home university tuition cost (ex. in my case, Swedish universities are free so I didn't have to pay tuition to my Japanese exchange school either). If you're a gymnasium/highschool student, you might have to pay a "program fee" (the money goes to the group organizing your exchange for you; if you stay with a host family, the host family might get paid a bit of it to cover your expenses with them). Usually you need to have studied to JLPT N4 (= Genki I and II) before you're allowed to go on exchange.

Entirely depending on your school, you can be in a host family or not. I've heard of university students who got host families but my Japanese school doesn't offer that.

2. Enroll in a language school. These are typically 2-year schools meant to get you to pass JLPT N1 and you have to pay some thousands of USD in tuition. I think this is the worst choice you can make (if you can make any choice). You're paying a ton more money than option #3 and getting the same (or less) stuff out of your time.

Anyone in Japan who needs proof of your Japanese level needs, instead of a diploma from a language school, a "you passed the JLPT" certificate or a university degree in Japanese. Having passed in a language school will not in any way let you SKIP low-level Japanese classes in university if you do decide to get a degree later (which a lot of people do, as most full-time jobs that'll make enough for you to pay rent require some kind of degree just to prove you're not an illiterate hick, and it's just easy to study what you already know). At my Swedish uni only a JLPT certificate lets you skip classes; not even my Japanese credits from another university transferred over. Many language schools probably require basic Japanese before you can enroll, but I dunno.

Instead of going to a language school (speaking as someone who planned to do this when they were in middle school) just start studying Japanese as much and as early as you can, take the JLPT in your home country and pass N2 or N1 by yourself; you'll get the same proof of knowledge while saving a crap-ton of money. Depending on how hard you self-study you can start from zero and get to N2 in ONE YEAR with not much of a problem.

A university degree in Japanese language studies is generally seen as more "impressive/worthy" and can get you more jobs than a JLPT certificate, according to my teachers. Job ads ask for JLPT certificates or "proof of equivalent knowledge".

3. You can contact a Japanese school you want to go to by yourself, then contact your own school in your own country by yourself, and arrange either an exchange or (if you're a university student) what's called a "research student" (研究生) status for yourself. Research students are as far as I can tell just a more unofficial exchange student. They can take all the same classes as exchange students.

If you do it this way and DON'T go through an exchange program, you won't pay your home country's tuition but you'll pay the Japanese school's tuition, whatever it is — meaning, for example, if the Japanese school is free then your studies will also be free. This method is how I moved from the USA to Iceland and studied in University for free (Sweden's also free but only for European citizens/spouse-permit holders). The catch is, depending on the school, you might not get the same help as exchange students get (if you can't read your bills you can't ask your teachers for help, you can't go on trips with the exchange students, etc).

What I found out today is, my school (宮城教育大学) has discount tuition rates for research students compared to say a japanese person going to the same school, meaning you only pay something like 30,000 yen a month in tuition. In order to get accepted as a research student you need to have taken the JLPT and/or TOEIC; what level of passing is required I don't know (probably just JLPT N3 though) BUT it's possible both those can be waived if you've been to that Japanese school before as an exchange student.

So my plan from now on is, if I can't find a full-time job and switch to a work VISA by the end of my exchange then I'll instead just extend my exchange studies at my Japanese school as a "research student", and that way I won't have to leave Japan while I finish my degree. One of my classmate's going to do that and that's how I found out this info.

If you can get a job and stay in Japan without a degree, proven Japanese knowledge, whatever, DO THAT!! If your uncle owns a takoyaki business in Osaka and will get you a VISA no problem, go for it!! But this is for us other folks, sigh.

If you're curious about what getting a 3-year university degree (= BA) in Japanese is like, look at my page here.