Sharp pain in lower right side but down near hip, close to pubic region. More likely ovarian cyst or appendicitis?

Hard to say. Nauseated? Pain duration? Quality of pain?Fever? Time of month? You should see a doctor for an exam and lab work to be safe.
Could be one of it. Some of the symptoms are common to both , beside other causes , fortunately ct scan will distinguish one from other, avoids surgery once appendicitis is ruled out, see your doctor or go to er.