Hey OP, my neck and that artery (left side) get mega-tight on me too. It's so annoying! I've been told that tight neck is a lot about stress posture and shallow breathing, so the tips to focus of relaxation should help in both ways. Dedicated deep breathing time everyday and continually check on your breathing to catch your shallow breathing and just might be helpful. I struggle with it, but people swear by it.
I also stretch my neck out as much as possible which is heaps painful, but does seem to help.