Extending residence permission, etc

I'm serious need for help folks! I want to study in Japan starting next April but I wish to move in earlier, at around July 12. There are several reasons for this, namely the fact I need to apply for the EJU (College test) till July 25 and, more importantly, the fact I want to be with my lovely girlfriend, whom I miss terribly.

So my question is this: if I go to Japan on July 12/13/14/15, around that time, can I obtain an alien registration card? I'm not a citizen of any of the countries which allow me to extend my stay, so I dunno what to do. Could I extend it by going to say, South Korea? Would I have to take a plane, or are there people checking your passport in Fukuoka (It'd go from Fukuoka to Pusan and back, ideally)?

Furthermore, on Japan-guide it says ''It is possible to change your status of residence (e.g. from temporary visitor to instructor or from student to engineer) at an immigration bureau inside Japan.'' This is slightly confusing because on the 'Students' section of the same page it says ''[...]need to get a student visa at a Japanese embassy or consulate outside of Japan in order to enter the country on a status of residence that permits long term studies.'' - Is this simply a matter of entering the country or do I have to go back to my country of origin before I can get a student visa?

In a nutshell (I apologize for this terrible mess, but I'm in desperate need of help and missing my sweetheart too much to express myself in a more articulate manner) I would like to know if I can someone extend my stay in Japan without having to leave the country, namely obtaining a gaijin card; and also if I can change my status to student after (hopefully) passing my exam, again, without having to leave the country.

INOH,

Alien registration is a formality for all non-Japanese nationals residing in the country for more than 90 days. It is not a requirement if you stay for shorter periods, and having an alien registration card does not entitle you to stay for periods longer than your visa, so it is probably not really relevant in your case.
If you want to extend your stay in Japan, then you will have to leave the country (e.g. to Korea) and return on a new tourist visa. Note that you are likely to be asked more questions if you appear to be doing a "visa run", and there is always the possibility that you could be denied entry if it is suspected that you are in the country for work purposes.