Sorry about the late reply. "Very affordable" was not the right choice of words. It's certainly cheaper than most colleges, but it may be too high, depending on your budget. Anyways, it varies by school and there are a TON of language schools for foreigners.

Here are the prices they gave me for the Intercultural Institute of Japan, at their Iidabashi or Akihabara centres, both in Tokyo. Tuition is ¥850,000/year ($7,800) plus textbooks and accommodation.

http://www.incul.com/eng/japanese_school/

Accommodation can be in the school dorms, at ¥67,000/month for 2 people, utilities included ($310/month each.) Or you can rent a single-room apartment in Tokyo, which starts at ¥60,000 ($550) plus utilities (water, power...) Finally there's the home-stay option, which costs around ¥5,000 ($46) per night, including breakfast, dinner, and otherwise living with a Japanese family.

All things considered, Tokyo can be quite expensive, so if you are on a budget, choosing a school in a minor town and only visiting Tokyo on the weekends may be the best option. Japan has excellent public transport, as you probably know.

Some people choose to stay in the campus dorms during the week and at their Japanese family's during weekends, to avoid commuting too much, while still getting a glimpse of the Japanese lifestyle. Others choose a school and dorms in one town, and a Japanese host family in another town, so they can study in Tokyo, for example, but have a place to stay in Kyoto during weekends and holidays. If you are below 30 you also have the option of Borderless Tokyo share houses:

http://borderless-tokyo.com/

In my case, I'm in my late 30s, so this entire thing is already crazy enough! I'm not sure home-staying at a Japanese family would be appropriate. I'll have to talk with my school about that. If it's OK, then I may try it. Otherwise, I'll start with the dorms and later on move to an apartment.

Another well known school is ISI - International Study Institute, with a big campus in Takadanobaba (Tokyo) and centres in Ikebukuro (Tokyo), Kyoto, and Nagano. This may be a more serious school and university? I'm not sure. You can see the fees on their website, they are similar to the ones for Intercultural:

http://www.isi-education.com/

Then there's Lexis Japan, with a modern campus in Kobe:

http://www.lexisjapan.com/

And then there's Kyoto Minsai Japanese Language School; Arc Academy at Tokyo, Osaka, and Kyoto; and so many more!

Choosing a school is really overwhelming, considering it will dictate how much you spend, where you live for the next year or two, how well you'll learn the language, what people you'll meet and what options it will open for you once school is over.

Later today I'll be speaking with someone from my local school (I've been taking Japanese classes at night) who manages visas and applications to these schools, so I'll probably have more info and will hopefully have decided where to apply myself!

School terms start in April, but most of them also accept enrollment in July, October, and January. They have students of all levels, so they will test you and place you in a class appropriate to yours.

Be mindful that you have to start doing visa paperwork at least 6 months before your departure. I hope I can still make it for July. Among other things, you have to show the Japanese government your bank statement, to prove that you can afford the school tuition and living expenses for the duration of the course; proof of school enrollment or occupation for the past year, to show that you have not been unemployed; and your high school or higher diploma. Other than that, I don't think they have any age restrictions.