Usually tourist visas in cant be extended in Japan.
You would need to leave Japan and come back with another one (or get another one on arrival, depending on which country you come from).

Other visas can usually be extended if you have a valid reason to stay in Japan (usually a continuation of the purpose of the original visa i.e. work, study etc )

You would need full documentation to support the extension - usually similar to what was required to get the original.

Usually you can apply for a visa extension up to 60 days before the expiration of the visa.

Applications need to be made in person at a Japanese Immigration Center.

What would be a viable way to leave Japan after 90 days if you intend to return? Would it be cheap to get a same-day round-trip to Korea for example? Sure don't want to spend the trip to Europe and back...

I hope I will get my student visa in time, just trying to cover some options.

Mike

Yes.. the old day trip to Korea used to work.. and sometimes still does.. but they are really on to it now
If there is any way to get your student visa sorted before you come and start on that.. it would be less risk.. and cheaper than a trip to Korea