Can shortness of breath occur from a muscle strain in my ribcage from weight-lifting?

Yes. If it hurts to breathe, we become much more aware of our breathing and this translates into a feeling of shortness of breath. Try some ibuprofen. If the pain improves and if you become distracted, the sensation should disappear. If it doesn't, you should be examined for other causes - eg collapsed lung (which is rare but does happen sometimes).