I logged on to ask the exact same question - great minds think alike and all that!
My circumstances are slightly different, in that a friend and I are starting the Camino Portugues from Porto next week and while we have our pilgrim passports already sorted out, we may be joined by a friend in Santiago who will then walk with us to Finisterre and (maybe) Muxia. Obviously, we'd all like to be able to stay in the same albergues once we walk on from Santiago, so I'm looking to clarify whether it's easy for someone to acquire a credencial in Santiago to walk the Camino Finisterre. I don't expect it will be a problem, but it's always a good idea to ask questions here and benefit from the knowledge of others.
Having walked the
Camino Frances last year (and then going on to Finisterre and Muxia), I'm guessing the pilgrim office near the Cathedral will be the place to go. I didn't need to acquire a fresh passport for walking on, but I would be surprised if that wasn't the place to go. If you're starting in Santiago, my only advice on that front would be to try to call to the office on the day in question before the flow of arriving pilgrims begins. The pilgrim office tends to the first stop for arriving pilgrims and from mid-morning onwards, that will be a steady flow, which can translate into lengthy queues!
As for your question about certificates/stamps once in Finisterre, you will get your passport stamped along the way as you walk from Santiago, and when you arrive in Finisterre (or at least when I did last June), you call to the Xunta (whether you're going to stay there for the night or not), present your credencial/passport, and they will present you with the certificate. If you walk on Muxia (or go there before Finisterre) you call to the tourist information office and they'll present you with the cert for reaching Muxia there.
Hope that helps, and I hope someone else answers the first part of your query!