Get checked. Could be carotidynia or a carotid artery dissection.
Nerve pain. Cannot imagine that pain goes through an artery, though nerves do travel in similar paths. Possibly a trapped nerve in your upper back is causing the shooting pain. Seek pain management advice, or neurology. My be that a steroid injection (seek a physician who is comfortable with neck injections - not all are) would help, or you may need surgery as a last resort. An MRI may be in your future..