Question: if we determine time based on the sun's position in the sky, what time is it on the surface of the sun?
Some time. Time depends on how fast are you going. Now the sun goes as relative to the earth as the earth is relative to the sun, so it depends where you are. For example in israel it is 12:00, but in the US it is 6:00. We see that the time is based on where you are.