Jeju is beautiful, BUT if you are alone, it is IMPOSSIBLE without a rental car. You can only rent a car with an international driver's license. You can hire a taxi by the day, but it's 100,000W. We used a Korean website, tbus.co.kr. If you go through an English speaker at the hotel, we were quoted 150,000 per day.

There is a bus system, but it doesn't work AT ALL to see many things in short trips. You'd have to stick to one place, like Seogwipo. Limo bus there from the airport, taxi around the area, then limo bus back to the airport. Jeju City doesn't have much at all. All of the sites are scattered around, mostly, the east and south of the island.

If you need to stay on the peninsula, try Gyeongju. It's 2 hours on KTX from Seoul. It's the old historical capital. Very quiet, but quaint, nestlted in the mountains. You can rent a bike for 10,000W a day.

Busan is also great and only maybe 15 minutes further on KTX. The only "problem" with Busan is that Haeundae/Centum City is around 40 minutes east of Busan Station by subway/bus. The other popular areas, Nampo-dong and Jalgachi, are very close to Busan station, but in the opposite direction (10 minutes tops by bus/subway). You'd have to make a solid plan of attack to see everything. Try Haeundae/Centum City the first full day and with your hotel. Then you can go back into Busan, explore Nampo/Jalgachi in an afternoon, catch the KTX back to Seoul.

I just got back from a similar Busan trip today. We left Seoul at 9:45 AM yesterday (Sat) at 7:30 PM tonight (Sun). There are trains continuously, so leaving earlier isn't a problem.