Is there a bus or shuttle service from Shenyang airport to Panjin city? If yes, how can I get it?

Yes, there are two buses from the airport to Panjin and the buses are at 14:00, 17:00, but I am not sure whether the time is suitable for you.

The buses can be taken at outside the Gate 8, 10, 12, 14.

If your time is not suitable, you can take the airport shuttle bus to reach SY Bus Station (SK Station) where you can find more frequent buses for Panjin.

Hope it helps ~