Gyeongju is easiest without a car when the historic centre is one walking block and bus trips are reserved for the longer connections. Do not use a bus map to recreate a driving itinerary stop for stop.
Keep the central sites together
Daereungwon, Cheomseongdae and Hwangnidan-gil sit around the same old-town area; walk this section instead of returning to a bus for each stop. For a first day, choose one continuous route and leave the distant temple or lake area for another transfer.
Use bus 10 for the terminal-to-Bulguksa line
Gyeongju City's route listing places bus 10 between the express/intercity terminal, central stops and Bulguksa. It lists a first departure at 06:00, a last at 21:40 from the terminal end and roughly 20-minute intervals. The city route page also advises checking live arrival information. A single line will not solve every connection; compare the actual stop and return time before adding a second district.
Place the hotel around the morning and evening
A central base keeps the old-town walk close. A station-side room makes an early rail departure simpler but adds a local ride to the historic area. Save the Korean address and entrance pin, then check luggage storage and the final bus or taxi for your arrival time. For the intercity leg, see the Busan–Gyeongju transfer guide.