Both are possible. Both arthritis and carpal tunnel syndrome (cts) can cause achiness in the fingers. Cts usually causes numbness and tingling in the fingers, which is worse at night. Arthritis is usually associated with joint stiffness, especially in the morning. It's best to see on orthopaedic hand surgeon for an evaluation.
Either or both. Carpal tunnel syndrome results from entrapment of the median nerve at the wrist and can cause pain, numbness and tingling (fingers falling asleep), weakness, radiating pain, etc. Many people with cts also have basal joint arthritis in the thumb (34% in a recent study). See a hand specialist; left untreated carpal tunnel can cause permanent nerve damage. Also, you could have neither one! good luck!