So I've been doing push-ups lately and I've noticed a sharp shock sensation up my arm sometimes. It's very annoying. I am healthy too. What is this??

Strain vs sprain. It's very possible that you strained a muscle and/or ligament OR sprained a ligament by overdoing things or perhaps from poor technique. Use RICE (rest, ice, compression & elevation) for now plus OTC pain relief eg ibuprofen OR naproxen and/or acetaminophen. See your FamilyDoc if no better in a few days to weeks, sooner if numbness/tingling/weakness or redness/swelling.