The space of JEJU ISLAND is 1,848 square KM. (FYI, the space of SEOUL is 605 square KM and the space of SINGAPORE is 719 square KM according to https://www.wikipedia.org.)
JEJU ISLAND consists of two cities; JEJU and SEOGWIPO. So, the north-half is JEJU CITY and the south-half is SEOGWIPO CITY. But, the majority of both cities are scenery of a countryside. I.e., JEJU CITY is condensed residential area at north coast and SEOGWIPO CITY is the other city-scape place at south coast.
Therefore, if your hotel is within JEJU CITY, you can reach your hotel by public transportation such as taxi, or bus.
But, if you hotel is outside JEJU CITY, you might consider to use private transportation.