Does spinal stenosis cause pain in the arms and shoulders?

Spinal stenosis. If it is lumbar and not cervical spinal stenosis pain in the upper body would be very uncommon, much more common in the legs and feet if it is lumbar stenosis.
It can. Stenosis of the cervical spine (in the neck) can cause spinal cord or nerve root compression which in turn can cause pain in the arms and shoulders.
Potentially. Stenosis in the neck can definitely cause pai in the arms and shoulders, there can be other causes as well,