Can i apply cortizone (hydrocortisone) ointment to the injection site of a flu shot if i have pain, redness, and swelling in the area? is that a bad idea?

Probably correct: Steroids suppress immune response & constrict local blood flow. The whole purpose of vaccines is to catch the immune systems attention to induce a strong response prior to exposure to the actual virus or bacteria. Thus suppressing local response could decrease the protective response to the vaccine. I do not think any explicit double-blind prospective controlled trials exist, too much$ for ↓value.