According to Wikivoyage and several other sources, Jeju Airport Limousine runs from the airport to Seogwipo, with some ten stops on the way. Is it possible to get on or off this bus at any stop? For example, could I travel from Jungmun to Gangjeong or Seogwipo? Or is this bus only for travelling to and from the airport?
The Airport Limousine (route #600) bus runs every 18-20 minutes between Seogwipo and the airport in Jeju City express, stopping only at a few select stops, including Jungmun Tourist Resort Complex (and International Convention Center), World Cup Stadium and terminating at the Seogwipo KAL Hotel.
These are all usable stops. They're there for people at the stadium/hotel/resorts to get on the airport limo to the airport, or to get from the airport back to any point. However, if you're wanting to use it to get from say, stop 2 to stop 3, there's no reason for a driver to say 'no you can't get off'. (That would be, in essence, kidnapping ;))
You may find, however, that the driver gets frustrated as there's no way to charge you a partial ticket, but if you're ok paying the full fare, just say you're buying a ticket, and then request an early stop.
It probably wouldn't be the most common, or economical use of the bus, however, and it may be worth investigating whether a taxi or regular city bus isn't more useful for short distances, depending on your planned route.