Symptoms such as you describe (swelling, red joint with fever) may be an indication of a systemic infection that requires immediate medical attention. Any opening in the skin causes a small injury and infection of the skin over the bursa can then spread down into the olecranon bursa.
A bursa is a sac made of thin, slippery tissue. Bursae (plural) occur in the body wherever skin, muscles, or tendons need to slide over bone. Bursae are lubricated with a small amount of fluid inside that helps reduce friction from the sliding parts. The olecranon bursa allows the elbow to bend and straighten freely underneath the skin.
The olecranon bursa can also become infected or septic. The sack fills with pus, and the area around the bursa becomes hot, red, and very tender. If the skin is infected too (a condition known as cellulitis), fever, chills, low blood pressure, and even confusion and other signs of mental impairment can develop.
With the history and type of symptoms you described, it might be best to seek medical attention first. A delay of even 24 to 48 hours while applying a home treatment could mean surgery instead of a simple antibiotic.