[Sangetsu]

No letter of apology will work, if your visa expires, you are screwed. With any luck, your passport received an "in-process" stamp of some type when your school submitted the paperwork for the visa.

Your ward office will probably not be able to give you much advice, more than likely they'll simply tell you to go to the immigration office. If you are anywhere near Tokyo, the main office in Shinagawa is the quickest and best place to go.

You should never have a third party submit/process your paperwork (unless you are outside Japan and overseas at the time). Get your school, employer, or whatever to give you the necessary forms/tax records, and take them to immigration yourself. That way you won't be apart from your passport for even one day. They'll put a stamp in your passport saying that your visa paperwork is being processed, and you won't have any doubts.

If you are walking around Japan now with no passport and no gaijin card, you can be detained by the police and held until someone can bring one or other to the police station (or immigration detention center). If they don't bring the paperwork quickly enough, you might find yourself deported.