When you apply for a spouse resident status, you should automatically get a 2 MONTH EXTENSION on your permission to stay in Japan. So no, 1 week isn't enough time to apply, but it won't be 1 week, it SHOULD be 2 months once you turn in the paperwork. As I said, you should confirm that point with immigration, but that's how its been every time I've applied regardless of how much time I had left on my current status. Now if you mean it'll take you longer than 1 week to get the paperwork and get to immigration, then yes, that is a problem.
A COE is a Certificate of Eligibility to enter Japan (
http://www.immi-moj.go.jp/english/tetuduki/kanri/shyorui/01.html).
Basically, there are 2 ways to get a legitimate resident status to stay in Japan longer than a 3 month tourist status. The first is to apply outside of Japan through your closest embassy, which is what the COE is. Once you've turned in the paperwork, you wait outside of Japan, and then from the embassy you're granted permission to enter Japan with a visa, which is exchanged for a resident card at the airport. You don't come back to Japan until you get that permission.
The second way is if you are already IN Japan, to change your status from tourist status to Spouse of Japanese National. As I said above, you'd go to immigration with all the paperwork, and upon acceptance of your application, you should be granted a 2 month extension from the time of application that you're allowed to stay in Japan. This is true whether you apply within the week, or you leave the country, come back to Japan, and apply with 3 months left on your tourist status. It doesn't matter which you do, as the end result is the same; within the 2 months (if your marriage is legitimate and all the paperwork correct), you should receive a 1 year permission to stay in Japan resident status. That's why I recommend getting the paperwork in NOW, as it removes the need to leave and reenter Japan, which could raise the suspicions of immigration.
Also, @Guestastic
The success rate of coming as tourist visa to spouse is 0%Frankly this is completely wrong. Immigration WANTS people to go the COE route as it gets them out of the country until they've got a legitimate status, but changing your resident status from tourist to spouse, or tourist to work, happens all the time.
Yes, its impossible to get a VISA from inside Japan, because a VISA is the document that allows entry into a country. What OP wants isn't a VISA, it is a Resident Status, which you CAN change within Japan. The problem is the terminology is used interchangeably, so its confusing.
tl;dr: OP, go to immigration immediately with all your paperwork to apply for the Change of Status:
http://www.immi-moj.go.jp/english/tetuduki/kanri/shyorui/02.htmlYou'll need to have your husband collect a number of documents and fill out some forms in Japanese, so I recommend you go over the necessary documents with him, or have him check in Japanese to make sure you've got everything. If you absolutely can't get the paperwork together before your status expires, then you'll have to make a decision of what to do. Just note, you'll STILL need all that paperwork no matter what, so do your best to collect it all ASAP.