If you don't mind spending about $10-15, a cab. We took a cab from the Swan to Poly for an early morning ADR. It worked very well. No muss, no fuss, no worrying about timing transporation.
As an alternative, you can take the bus from the Swan to MK and then either walk or take the resort monorail one stop to the Contemporary.
As a more scenic alternative involving a bit more walking, you could walk/boat to Epcot international gateway, exit at front gates to monorail to TTC and take the resort monorail to the Contemporary.
Resort to resort transport can be a bit tricky with Disney transportation.
