I stayed in Tokyo for almost a year, I did a homestay. There is websites for people to look for people, I personally found my familly trough a penpal friend. I know that there is also schools that can refer to a familly (or place you with a familly), when you are one of their student.
The advantage is that if you live with a Japanese familly, they can help you doing some administrative things. My familly helped me to get my alien card (when it was still a thing), and to open a bank account. An other advantage could be to practive your japanese with japanese people (instead of living wiht other foreigners in a shared house), however, my familly was always hosting 4 foreigners, that was definitively a fun experience, but of course I did not used japanese that much at home.
For price, I paid 2500 a day (that was 5 years ago), that include everything that you would have to pay yourself in an apartment (gaz, electricity, water, invernet), and it also included 2 meals a day.
You can find different type of familly, some prefer short term other long therm, some female only, some male only.
I just found my familly on that website : https://www.homestay.com/japan/ so it looks like a place to check if you like this idea.