Is it possible for people with polymyalgia rheumatica to have sharp pains in all the finger bones?

Yes: People with polymyalgia rheumatica can develop hand pain. Some people with polymyalgia rheumatica also go on to develop rheumatoid arthritis, which causes hand pain.

Polymyalgia rheuma: Polymyalgia rheumatica is an inflammatory disorder that causes muscle pain and stiffness. The pain and stiffness often occur in your shoulders, neck, upper arms and hips. You might have pain in your hands and fingers as well.