The medical term for your condition is cervical radiculopathy from degenerative disease. That’s a mouthful to really describe a pinched nerve from age-related changes in the spine. Just as you described, cervical radiculopathy can develop as the discs thin out, lose fluid, and compress down. Just that change alone results in a series of other changes as well.
Disc disease leads to an increase in the pressure and load on the vertebrae. It also means the facet (spinal) joints move closer together. Compression and added load there can cause bone spurs to form as the body responds to the added friction and shear forces.
A short space between the vertebrae also affects the spinal ligaments with resultant increase in stiffness of the spine. All of these factors together reduce the size of the foramen (hole through which the spinal nerve travels as it leaves the spinal cord and travels down the arm). And over time, with thinning of the bones and compression, the front of the vertebral bodies get pushed down. Vertebral compression fractures can develop. These are two more anatomical factors that can contribute to narrowing of the foramen.
The final result? Pinching or pressing on the spinal nerve root(s) and cervical radiculopathy. Studies show that in the case of a simple nerve compression, the condition is self-limiting. This means it will probably get better on its own without treatment. The length of time it takes for this to happen varies from person-to-person.
The physician who examined you and made the diagnosis may have some suggestions for you based on the location of the spinal changes and severity of your symptoms. If the painful symptoms get to be too much for you, nonoperative care may offer you some relief. This could include nonsteroidal antiinflammatory medications, massage, physical therapy, and/or chiropractic care.
You may be able to find something that fits your budget and helps in the early (acute) stages of the problem. Hands-on therapy may help improve alignment and reduce pressure on the nerve. Antiinflammatories reduce swelling around the nerve and can also result in pain relief. Combined together, the medications and the manual therapy could help you get through to complete resolution of the problem.