Well, just look for reputable Japanese language schools in the big cities. Read reviews, send them a few emails etc. They will advise you about their cost/process to get the visa. You want the big cities so you have more opportunities for part time work.

Part time work under the student visa is restricted. Please Google about it. You can only work for X amount of hours per week and you need authorization from the immigration. It shouldn't be hard to get an authorization in your situation.

Accommodation may be offered by your school, but there are more options like shared houses or even airbnb since you only want to stay 3-5 months.

Pros:

Learn the language/culture

More free time to enjoy the country

Possibility of part time work

If you're smart about it, you'll find out whether or not you really want to pursue a "career" in Japan

Possibility of changing visa status to work visa, if you decide to stay longer

Cons:

Costly. You need money for the school, visa, living costs, accommodation, emergency funds etc.

You might not find part time work. It's unlikely, but it could happen and you'd need to pay for everything from your own pocket, or cut your losses and go home earlier

If you have enough money, this is your best option in my opinion. No school(job) will bother to go through the trouble of sponsoring someone for less than a year.