Can I stay after buying a house in Japan?

There is no saying that you can immigrate to Japan without buying a house. The correct statement is to obtain a business administration visa (a long-term visa) by setting up a company in Japan. If the company is a real estate investment company and can get the real estate license, then "buying a house+real estate company = immigration" can be realized.

Because Japan is cautious about immigrants, only by holding a long-term visa and meeting the corresponding requirements can it obtain naturalization or permanent residence status. Long-term visas to support foreigners to stay include business management visas, work visas and marriage visas. If you want to do a good job in immigration or permanent residence, you need to plan ahead what kind of long-term visa you choose to live in Japan according to your own situation.

Generally speaking, permanent residents, special permanent residents, ordinary permanent residents and permanent residents will stay in Japan for a long time after obtaining the corresponding visas, including naturalization by changing their nationality.

Fixed residence visas are mainly made up of Japanese spouses, or spouses of people who hold Japanese permanent residence visas, or Japanese orphans who come to Japan, or minors who come to Japan with their parents and get married in Japan. Like work visas and study visas, they need to renew after their stay. The term of a fixed visa is generally three years, and it needs to be renewed after the expiration. You can apply for a permanent residence visa if you get a second three-year visa.

Permanent visa, with permanent residency in Japan. Under special circumstances such as not violating Japanese laws, you can live permanently in Japan.

References:

Baidu Encyclopedia-Japanese Settlement Conditions