Thanks to the convenient transportation in Ulsan, people can travel easily over the country, such as Jeju Island. 
Jeju Island is the biggest island in Korea, which is one of the List of World Heritage Sites. It is the warmest place in Korea. Every year, millions of tourists are attracted by the stunning scenery. Also the Jeju Airport is the third biggest city in Korea, receiving over 20,000,000 a year around from home and abroad. It is convenient for people get from Ulsan to Jeju Island. Here is the recommended route from Ulsan to Jeju Island Airport.

Airplane

Ulsan Airport is an airport in Ulsan, South Korea. According to the survey in 2011, over 550,000 passengers used the airport.
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There are 2 direct flights from Ulsan Airport to Jeju Airport every week.
Depature Arrival Duration Date
13:35 14:35 1 hour every Fridays
16:35 17:35 1 hour  every Sundays
(The timetable is used for reference)
Then you will arrive at the Jeju Airport after a hour flight.
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Ferry

If you want to go from Ulsan to Jeju by ferry, it would take quite a long time to get there. But the great scenery along the way worth the pay.
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In Ulsan, you go by light railway to the Daegu Geumho Bus Terminal and transfer a long-distance bus to Wando. Then you can go to the Jeju Island by ferry, which takes nearly 3 hours to get there.
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Some Tips for taking the ferry :
1.Please check the departure schedule and fee with the ferry cooperation before departure due to weather and other conditions.
2.Please bring your ID card at boarding, and finish boarding 10 min. prior to departure.
3.The ferry fare will increase by 10 percent during specific periods, such as summer vacation, national thanksgiving and Korean New Year holidays.