Swapping from work to spousal visa and changing last name

So the main process I am pretty clear on: go to city hall, sign the documents to make the marriage official, update the family registry, go to immigration and bring the usual visa stuff + marriage certificate, family registry, and the questionnaire about the relationship. This is all pretty clear.

What I’m curious about is the name changing process. In a perfect situation, since my fiancé is super busy, I’d like to do the city hall signing in the morning and go to immigration the same or next day. However do I need to update my passport to my new name ? What’s the sequence of events for that ? Can I take his last name on my zairyu card and THEN my passport ? Or do I have to do all the work with the embassy to change my name and then apply for the visa ? The immigration website seems to not specify about the passport name which is why I’m confused.

Thanks in advanced