I did a little bit of research for you. The Brachial Plexus Block is a common procedure for local anesthesia for surgery (performed by an anesthesiologist). After reading about it, I'm pretty sure I had this anesthetic block when I had hand surgery to repair a ruptured ligament (basketball injury) way back in high school. It just numbed my entire arm so it must be different than the blocks used to treat chronic pain.

I did find some information regarding brachial plexus blocks being used for chronic pain conditions:
http://radiology.rsna.org/content/216/3/886.full
http://www.deaconess.com/body.cfm?id=1918

If Dr. Jordan is performing this procedure, I would guess he is using ultrasound guidance which I believe is superior to x-ray/CT guidance mentioned in the articles. Regardless, I'm not sure I would personally opt to have this procedure since it will only provide you with temporary relief and this is such a sensitive area which could lead to complications. It scares me a little, kinda like Dr. Ahn's angiogram/angioplasty.

I can recommend the cervical facet blocks without reservation. I think it is important for you to identify what pain is originating from the C-Spine. TOS is often diagnosis of elimination, while spinal issues are relatively easily diagnosed. You very well may be suffering from double crush syndrome which I was. The facet block was scary for me the first time around just because of the unknown. I have had it done 3 times cervically and once in the lumbar (I have bad discs down low too), in addition to other epidural injections. After the first couple times, I am now able to undergo the procedures with no sedation! It was after my 3rd cervical facet block (and multiple MRIs) that we (neurologist, neurosurgeon, my wife, and me) concluded that my cervical fusion was fully healed and successful and the remainder of my symptoms must be due to TOS.