Yeha Guesthouse; where I spent the nights while I was in Jeju-si was 5 minutes walk from Jeju Local Bus Terminal. There's a bus (number 300) running to and fro the airport from Jeju-si approximately an hour for 900 won. If you don't have the time to wait for the bus or in a rush, just hail a metered-taxi for about 5,000 won or less. The airport is about 10 minutes ride and taxi meter starts from 2,200 won.
Jeju-si in early morning.
If you walk out from Yeha Guesthouse, turn right and walk until you
see HyundaeSports shop on your right. You will be at T junction, and
cross the road.
Once you cross the road, just walk towards the main road and you
will see the traffic light, turn left.
Walk straight after passing a few shops and police station. Jeju Local
Bus Terminal is just next to it.
I paid 3,000 won to go Seogwipo-si and the bus runs every 10 minutes.
There are many bus to Seogwipo Local Bus Terminal only the route will
be different. If you are heading to somewhere else, make sure you check
which bus number.
FYI, You may use T-Money in Jeju Island.
The bus schedule to Seogwipo-si.
I didn't know that T-Money was accepted here so I bought a
ticket at the counter.
Some bus stops like this offer free Wi-Fi. Isn't it cool. I frequently
connected while I was at Seogwipo Rotary bus stop.