Gol Gul Sa Temple would not be something familiar to most Koreans and they probably would mistakenly thought you are talking about Bulguk Sa Temple. Golgulsa Temple is the only temple cave in South Korea and built completely out of stone back in the 6th century. I, hereby, sincerely suggest you should go to Gol Gul Sa Temple for a walk, though it would be very exhausting but it would not disappoint you.
GyeongJu is a place where you can find a lot of architecture and temples that were built hundreds of years ago (during the Silla Kingdom). Not only does it appeal to foreigners, Gyeongju is also popular among Koreans who are on a short weekend trip.
You can take bus #100/150 outside the train station, bus station of Gyeongju or near Anapji Pond.
The main attraction of Golgulsa is the sculpture of Maya Tathgata Buddha and the Gwaneum Cave. It would require you to climb up the hill (Yes, you read right, it is a climb not a walk. Technically, there are stairs but it’s about 80 to 90 degrees and you would have to hold on tightly to a rope.) The Maya Thatgata Buddha is around 4-meters tall. You can also see hundreds of little buddha stone carvings in different caves.
Golgulsa Temple also offers temple stay.
From Golgulsa to the underwater tomb of King Munmu would require another ride on bus #150 then get off at the beach. The underwater tomb of King Munmu is a little rock 20m off the shore. King Munmu was the 30th ruler of the Silla Kingdom and intended to protect the country such that he required his subordinates to bury him in the sea.
Traveling to GyeongJu:
1) Buses were a common and a cheap way of traveling around South Korea. It would be a 4-hour ride from Seoul to GyeongJu Bus station. There are also buses running until midnight both from and to GyeongJu with a (20%) higher fare.
2) Train. The lowest class of train will only cost you around 60% of the fare of E/M class bus. It would be around 4 hours to a 4-and-a-half-hour ride.
3) KTX. A new KTX station (Sin GyeongJu Station) was opened so people can easily travel to Gyeongju by KTX. It will only take you around 2 hours from Seoul to Sin GyeongJu by KTX. However, it would be the most expensive way of traveling.
[P.S. there are passes for foreigners traveling in South Korea for 1/3/5/7/10 days and you can take whichever class of train for an unlimited time.]