Could this wrist pain be arthritis or carpal tunnel syndrome?

Possibly. It could be either one of these but it may be other problems that your doc could help resolve. Remember pain in your body's way of telling you there is a problem. Don't ignore it if it persists .
Either / neither. Both arthritis and carpal tunnel cause wrist pain but so can fractures, tendinitis, cancer, gout and numerous other conditions. If severe or impairing, see your doctor for testing.
Possibly. Wrist pain can have many causes. Common causes of wrist pain include tendinitis, sprains, fractures, carpal tunnel, and arthritis. If your symptoms are severe, started after a trauma/fall, have not improved, or interfere with either your regular activities or your recreational pursuits, you should see an orthopedic surgeon to get properly diagnosed.