How do I heal a 7 year old cat that has an oozing sore near tail?

The most important is to clean and disinfect. You can use topical antibiotics as e.g. powders with a broad spectrum antibiotic but always first clean good without producing further damage. You can also drop some hydrogen peroxide of the type commonly used in wounds and if you prefer a natural approach, use Apple vinegar.

The second step is to protect it. There are some medical micropowder sprays that cover the wound with a thin layer of a metallic looking film that will prevent further infection and stimulates healing.