Well, you have 3 choices in getting to the Contemporary from Epcot and back. You can take the monorails, drive over yourself, or take the bus. Each has its disadvantages.
You can leave your car at Epcot and take the monorail to the TTC, then transfer to the resort monorail and get off at the Contemporary. This is probably the slowest method, but the most fun, especially if you ask to ride up front!
You can drive over to the Contemporary, which is the fastest, but you have to go through the far right lane of the MK entrance, and then tell the guard at the Cont. that you are eating at Chef Mickeys. Since you have already paid for parking at Epcot, you shouldn't have to pay at the Magic Kingdom, you'll just have to stop and show your receipt. If you are comfortable driving around WDW property, then this is the way to go.
I believe you can also leave your car at Epcot and take a bus over to the Contemporary, but that's not very exciting, and busses can be very slow, especially when you're trying to make an appointment. Either way, I'd give yourself about an hour to get there.
Good luck and have fun!