I have a couple of little manual laminators similar to this and they can work pretty OK.
Foam core can be tough with them but it can be done. I prefer pre-coated art board for the mounting. You are correct in that you need to laminate the print first. The biggest issue will be dirt ... any minor spec will leave a bump in the print.
To apply the laminate, I've gotten the best results using a carrier board. This can be made from a stout piece of plexiglass to which you adhere a piece of release paper. The release paper to make this requires it to be single sided release paper. To give you an idea of this, you would apply the laminate much like they apply the facemount adhesive in this video. http://www.youtube.com/watch?v=fcUgilc1mqU