I cannot give you a definite clinical diagnosis without seeing you.
TOS (or Thoracic Outlet syndrome) is a commonly overdiagnosed problem, and is actually very rare. This diagnosis should only be made by an experienced neuromuscular physician with the aid of an EMG
The presence of an extra cervical rib does not necessarily mean that it is causing the symptoms - this anatomical variation is relatively common in the general population
The test of choice would be an EMG and nerve conduction study which will detect any significant pinching of the nerves, or nerve roots. Your local neurology center should be able to do this. An MRI will also be useful to look at the discs and nerve foramina (the openings through which the nerves go through to go down the arm from the spinal canal)
Treatment will depend on the cause
Good Luck