Which application you need
Four routes, decided by where the foreign spouse currently is.
Outside Japan — Certificate of Eligibility (在留資格認定証明書交付申請). The Japanese spouse files it while the applicant is still abroad, at the bureau covering the address the couple intend to live at in Japan. No fee. The certificate is valid three months from issue, and the applicant must make a landing application within that time. Standard processing period one to three months. The ISA states that this application cannot be filed by post.
Already in Japan on another status — Change of Status (在留資格変更許可申請). Filed at the bureau covering the applicant's registered address. 6,000 yen on permission (5,500 yen online). Standard processing period one to two months.
Already a spouse-visa holder, expiring — Renewal (在留期間更新許可申請). A holder of a period of six months or more may file from about three months before expiry. Same fee. Standard processing period two weeks to one month.
A child born in Japan, or loss of Japanese nationality — Acquisition (在留資格取得許可申請). Filed within 30 days of the event; no fee.
Who may actually file is not the same on every route, and this catches people out. For a Certificate of Eligibility the law appoints a relative living in Japan as the applicant's 代理人 (representative), which is exactly how the Japanese spouse is able to file while the applicant is abroad. For a change of status or a renewal there is no such representative: the applicant must appear at the immigration office in person, and the Japanese spouse's role on those routes is guarantor. A spouse can accompany the applicant, but that is not the same as filing for them. The exceptions are an accredited lawyer or 行政書士 (administrative scrivener) acting as a filing agent, a legal representative, or a relative where the applicant is under 16 or cannot attend through illness.
One route that usually does not work. Changing from 短期滞在 (Temporary Visitor) to a spouse status inside Japan is not permitted unless there are compelling special grounds. Couples who marry while one is visiting on a tourist entry are normally expected to use the Certificate of Eligibility route instead.