It is most certainly possible to change visa types within Japan. As for getting married here, it is a terribly simple matter. Takes 10 minutes at city hall after you and she provide the necessary documents (also quite simple). She needs to provide a copy of her koseki (family register), and you probably need to provide some sort of document from your embassy confirming/swearing that you are currently single (check with your embassy). That, plus signatures from 2 witnesses, and you are in. You don't even get a marriage license unless you want to pay extra for it.
Ceremonies are the expensive thing, and you don't need one, but if you want one, that's a whole different ball game.
Once you are married, go to immigration and fill out the forms. You may have to provide a few photos of the two of you together, and perhaps the paperwork is longer for some now (just for security sake), but it's pretty straightforward. Even if you both are not working, you can get her parents to sign as guarantor, and you will probably get the spouse visa in 3-4 weeks.