Foreigners owning a business in Japan?

I am here on a contract and have a sponsored visa. I am also starting a business here with a Japanese person. When my contract is up will I be able to continue with my business, despite no longer being sponsored by my current company? Anyone have any idea of the legalities of this?

I assume you are employed by a company currently, which got you one of the work-permitting visas for the time being (such as Engineer, Specialist in Humanities/International Services, etc.), and that you do not have Permanent Resident or Spouse of Japanese national status.

If your current employer-sponsored visa expires, or actually before it is up, you will need to either (1) get your Japanese partner to incorporate the business in Japan, and have him sponsor you for the visa, meaning, your local partner being the employer and you the employee, or (2) register as a business yourself and apply for Investor/Business Manager visa status yourself, if you are the business owner.
http://www.mofa.go.jp/j_info/visit/visa/04.html

I believe there are some requirements for this Investor/Business Manager status, such as that you need to employ a certain number of local staff. I've seen other threads mentioning that this status is quite difficult to obtain.
http://www.mofa.go.jp/j_info/visit/visa/appendix1.html

Thank you for your answer. This confirms something I had heard before. Can anyone else confirm this.

get your Japanese partner to incorporate the business in Japan

I am not 100% sure on this, but I don't think an incorportaion is required. I think he should be able to sponsor you even if he were to run the business as an individual.