Tennis elbow isn't osteo arthritis, LJ.
Anyway, what can help is rest, ice at first, heat later, anti-inflammatories (like ibuprofen, aspirin, naproxen sodium), exercises after it heals to strengthen the muscles supporting the joint.
It isn't necessarily going to be a chronic condition if you've never had it before and it's obviously due to an injury. It might, might not. If it becomes a longer term problem, a doctor can give you specific exercises to do to help strengthen the arm, may give you prescription anti-inflammatories, and possibly cortisone injections.