Could the steroid injection get me rid of the pain in my left buttock for a long time.? Does it have a side effect especially that I have fibromyalgia?

Diagnosis? The answer is likely yes. It depends on the cause of the pain. Have you seen a doctor? You may have sacroiliac pain or piraformis syndrome.
Depends! If injected im, you were getting a quick dose of cortisone. Won't last. On the other side, if the muscular trigger point was injected, expect longer relief!