Is bursitis treatable with medication?

Yes. Most cases of bursitis are inflammatory, and thus can be treated with anti-inflammatories. This includes oral and injectable steroids as well as topical or oral nsaids. Infectious bursitis, called septic bursitis, requires use of antibiotics.
Yes. Nsaids (non-steroidal anti-inflammatory drugs) like Ibuprofen and Naproxen sodium are common medications used to treat bursitis. Heat application, reduced use of the affected area, and targeted exercises are also commonly used.