It'll be open at the normal Epcot opening time. Walk time from the Swan isn't that bad if there's not a boat waiting for you. With the boat - you will board it first, then have to go to the Yacht/Beach Club, then the Boardwalk, and then finally to Epcot. It's much faster to walk if you can. BTW - if you take the walk via the Boardwalk, it's about a minute faster than the way via the Yacht/Beach Club. Can you tell I've done this a lot?
Once you enter Epcot, the walkway towards France will be roped off and there will be a Cast Member standing there to direct you through an empty United Kingdom and then Canada. Lots of picture taking opportunites here without any crowds. Once at the walkway to Future World, you'll encounter another Cast Member who will direct you towards Future World. You won't be rushed along at all.