I think that your diagnosis is correct in that it does sound like an infected olecranon bursitis. This extremely painful and can quickly spread if untreated. However as he started a course of antibiotics this morning, he should see some improvement i his elbow within 48 hours.
Make sure that the pain is relieved by putting it in a sling and giving him ibuprofen 600mgms when and if necessary up to every six hours.
Keep us informed.