Hiya
If you wish to stay for two years, a student visa would be the best option. If you live in the UK it will be quite easy for you;
Embassy of Japan in the UK
If you live in a visa exempt country, then depending on which country you live, you can stay from 6 months to 90 days max;
Embassy of Japan in the UK
Anyway if you have a student visa, a spouse visa should be easy to aquire, as long as you can prove you have the funds to support yourself and your family. As for conducting your business online, I highly doubt you should even mention that in your application, it will be classed as working while on your visa and you will be refused for definate (just keep that info to yourself
).
If you got cash to burn exit and re-entry is a visa free method, but it is gonna cost a shit load of money compared to getting a student visa and returning home once every 6 months or so.
Well good luck to you and hope you find the best method for you and your family.