"There's a time and place for everything. And it's called college"
A priest of time's duties depends a lot on whether chronomancy, and time travel in general, are possible. If so, they look after the timestream, make sure noone's messing with history ("take nothing but looks, leave nothing but footprints"). They're here to ensure that things happen one after another.
Other than that, they would probably do temporal divinations. Looking into the past, the future. History would be a big thing for them, because it's so much to do about time. Telling the time, knowing the time.
Oh, and horology (that's nothing indecent). They wouldn't have simple sundials. They'd have some advanced stuff (magic, tech, pick two) to tell the time.
Know the time, tell the time, remember the time.
Now Space: That's all about travel, distance, going places. Teleport, flying, travel magic. If your campaign world's a Roundworld rather than a Discworld and has some Space around itself, they'll be into astrology (probably also if they're on a Discworld).