That's the easiest way. Take the bus directly from the airport. ICN airport would have much more regularly scheduled buses. A quick Google said that they come around every 15-30 minutes.
Incheon Airport is just 30-40 minutes further west of Gimpo Airport.
For general ease of flying, Gimpo is a simpler, faster and smaller airport. But fares are usually higher (caters to business traveler for short regional hops) and I am sure the bus there is much less frequent to Cheongju than Incheon Airport. I am bored at work and found it on the airport website, the bus time... and the Gimpo bus goes to North Cheongju Bus Terminal, not main one.
08:20 / 09:40 / 12:10 / 13:10 / 15:20 / 17:20 / 18:30 / 22:10
https://www.airport.co.kr/gimpoeng/e...04&menuId=1834
Cheongju has an airport that used to serve China, but those flights have all stopped since THAAD and tourism is down.
https://txbuse.t-money.co.kr/ is the site with English option to reserve your bus ticket from Incheon if you want to do it online. Depending on bags, immigration line, you can be out of the airport in 15 minutes or 40 minutes haha.