Can arthritis meds cause excessive sleepiness?

Not typically. Having said that, Indocin (indomethacin) which is an older arthritis med can definitely cause sleepiness but most of the other nonsteroidal anti-inflammatories do not generally cause sleepiness in my experience.
Usually not. Some people sleep better because they in less pain with nsaids. Most of the arthritis drugs do not cause drowsiness, except for drugs for spasm (soma, etc), flexeril, amitryptiline, and lyrica (pregabalin).