Gwangju Airport connects to Gwangju Metro Line 1 through Airport Station. Korea Airports Corporation says the station is about a 500 m walk from the terminal along the sidewalk. The airport's English transport page currently lists subway service from 05:50 to 22:40 and a full-line running time of about 38 minutes.
The airport currently lists an adult subway fare of KRW 1,400 for a single ride and KRW 1,250 with a transportation card. Because fares and operating hours can change, treat the airport page and the Gwangju Transit Corporation as the final check before travel.
A taxi can be the simpler choice for luggage, late arrival or a destination that is awkward by rail. The airport currently gives example taxi estimates of about 8 minutes/KRW 4,000 to Gwangju-Songjeong Station, 16 minutes/KRW 9,000 to Gwangju Central Bus Terminal (U-Square), and 30 minutes/KRW 15,000 to the Asia Culture Center. These are estimates, not fixed fares, and the airport warns that traffic can change the amount.
Do not search only for 'Gwangju' and assume every hub is interchangeable. Put the exact destination name into your map app or show it in Korean: Gwangju Airport, Gwangju-Songjeong Station, U-Square, or your downtown destination.
Practical next step: if your arrival is within metro hours and your destination is on Line 1, use Airport Station; otherwise compare a taxi to the exact hub you need rather than going to a generic 'Gwangju Station.'