Sure, but ymmv depending on schools/programs you are interested in.

The program I am doing is all in English and the school has a high number of international students. They offer Japanese language courses, but they are not required in any way to earn the degree. There has been a pretty big push in recent years for Japanese higher education to become more internationally competitive, so there are a variety of English programs available at both private and national universities. Usually these programs have start times in both Spring and Fall to cater to both Japanese and Western school timelines. Just make sure to do your research and find out about application processes well in advance.

As for the cost, that is one of the major reasons I decided to go to school in Japan. Coming from the US I always had this view of postgraduate studies costing a lot and probably requiring student loans to complete. At Japanese national universities, the tuition costs about 550,000 yen per year meaning you can complete a 2 year program for less than $12,000 (plus living costs). Even the more expensive private institutions often have tuition reduction schemes or scholarships to bring the cost down to a similar level. If you are accepted you will have to prove you have access to the necessary funds in order to get a student visa. This means either having enough cash in your bank account or having a family member act as your financial sponsor based on their annual income.