Sure you can. I'm doing it. I have been here in Japan for over 15 years. I have set my roots down deep and I have no plans to leave.
First of all, let's talk about your degree. Basically if you have a BA degree you can find work. It would be a lot better for you if your degree were in English, education, linguistics, or TESL/TESOL, etc. I have a friend who has an engineering degree and is here only because he got married to a Japanese girl. He hasn't had full-time work for about 10 years. He is doing the part-time thing, which is working at 10 different schools during the week. An hour here, and hour there. He is making a lot of money, but his schedule is killing him. One reason is that his degree doesn't support teaching English even though he is a native speaker.
Your lack of Japanese abilities is not a problem. It would help, but it is not a big deal.
Full-time jobs are at a premium in Japan, so basically the idea of working at different schools is not really plausible. If you are looking at private English schools, you could do it, but not at public institutions or institutions of higher learning.
As far as salary, you could start at 200,000 a month, then go as high as 500,000 a month in time. When I did the part time thing (for one and a half years) I was making around 350,000 a month but with potential for a lot more but the schedule would have been too hectic.
If you work full-time, you can get full benefits, and they are great. If not, you will have to purchase health insurance from the government, but the percentages are a lot higher than the company insurance. Retirement only comes from full-time positions and the benefits are compiled yearly. So the longer you work full-time, the more you get when you retire.
Bottom line, you can work and live forever in Japan. But where you work will play a large part in your quality of life.
It was easy for me because I had a PhD in Linguistics, and I am married to a Japanese national. I also paid my dues working the hard life path for 5 years.
Contact me if you have anymore specific questions.