There are no monorail stops "inside" the parks, so you can ride from any of the monorail stops without using an admission day on you park passes.The stations are outside Epcot, MK, the Ticket and Transportation Center, and at the three monorail resorts (Contemporary, Poly, and GF). There are actually two different monorail routes; you switch from one to the other at the TTC.
I suppose the closest monorail stop to the Swan is the one outside Epcot. Unfortunately, you cannot easily get there from the Swan without using park admission, since the normal way to get from the Swan to Epcot is to take the boat which drops you off at the International Gateway entrance to Epcot (located between the France and Great Britain attractions in the World Showcase section of Epcot) rather than at the main entrance to Future World where the monorail station is located. It would probably be quickest to take a bus to the MK and walk to the monorail station without going into the park itself. Although a greater distance, that bus trip is actually pretty quick, since it makes only one stop (at the Dolphin) and then heads on over to MK. It's probably a two-minute walk from the bus drop off to the monorail station.
Ralph