Can sleeping with the air conditioning blowing on you cause sore, aching shoulder and neck muscles?

Air conditioning: Not likely unless you are too cold during the night. Try to sleep without air conditioning a few nights and see if this helps. If not see your doctor about why your are having a problem.

Cold muscles: Absolutely... A cool stream of air, even it if feels nice, can cause stiff neck- and shoulder muscles and make you wake-up achy... try a fan or so with indirect flow of air directly to you as an alternative. Or wear a thin shirt with a collar maybe? Good luck!