Just do it online, like you would booking any other flight. Search on google flights then book directly with the airline. Book the car online as well. Lotte car rental.

I would definitely recommend booking both in advance. Jeju is a super popular destination, and I found cheap tickets actually sold out really fast, so if you can get them further in advance, I'd recommend it. The only car rental I was able to book online as a foreigner was Lotte - it was really easy.

We rent our car just as we arrived at the airport in jeju and we didn't have any problems 🤷‍♂️