Treat Gongju Station and Gongju's heritage center as two separate stages. Current public-transport data derived from national transport data shows route 200 linking Gongju Station with stops including Gongsanseong, Sanseong Market and Gongju Bus Terminal, but the exact schedule must be rechecked on the travel day.
For a first visit, decide which old-city stop you actually need. Gongsanseong, Sanseong Market and the Muryeong royal-tomb/hanok-museum area are not one station-sized cluster, even though they can be combined on foot once you are in the center. Save the Korean name of your first stop before leaving the station.
If your train arrives late, check the final bus from Gongju Station before boarding the KTX/SRT. Do not build a tight return connection around a static timetable. If the bus no longer fits, compare a taxi from the station rather than traveling first to the wrong terminal.
The useful rule is: long-distance rail to Gongju Station, then a separately verified local transfer to the exact heritage stop.