Name change on residence card

Hello,

I have a permanent visa status, but I changed my name after my marriage with my Japanese wife.
I have checked the appropriate procedure on the internet, but it seems they ask also for a marriage certificate showing the new name. My problem is that I changed my name in my home country after our marriage, but don't have a marriage certificate and in the koseki tohon it still shows my birth name.

I have now my passport from my home country showing my new name and birth name, is this alone not enough to re-issue my residence card?

Also how long does this procedure normally takes, I assume it should not be very complicated, so shouldn't take that long, but any advice would be appreciated.

Also I am about to move in 2 months and for that do I understand correctly that the notification of a new address is rather easy, where I just have to go to the city hall of my new city instead of going to the immigration office for the whole prefecture?