What helps me:

1. First I have bilateral TOS but pain on Left neck/shoulder.
2. Yoga. Chest openers (any). Roll shoulders back, pinch blades. No eagle pose, no down dog. Nothing that puts any body weight on shoulders. No push ups or plank.
3. Cardio exercise. Increase blood flow to muscles.
4. Long foam pillow against spine when sitting in car/airplane/in front of boob tube. Opens chest more.
5. Watch the posture in front of computer or when sitting. Remember...chest LIFTED, shoulders down and back. (no concave chest).
6. Watch neck posture. Lengthen back of neck, chin down or paralell and not sticking out front (no forward lean to head..it should be straigth up and down in line with backbone).

don't know about cervical rib...don't have that.
If elbow bent a 90 degrees while sitting...could be that you are applying pressure (even slight) to the shoulder...don't! take pressure OFF elbow.

7. Watch what you are "carrying" In my case my problem area is the "coracoid process" this is where the pec minor attaches as well as the bicep tendon (one of them). When carrying packages in front.these two suckers PULL down on the coracoid and create havoc for my Brachial Plexus underneath. One observation.