Angel, oophorectomy is actually REMOVAL of the ovary, which is kind of worst case scenario.
As for the cyst, most cysts are harmless. going on the pill can reduce the cyst and prevent new ones from forming. if it is a simple cyst, and pretty small, chances are they won't do anything about it. if it grows and starts to become solid, it's another issue. if you're worried, ask your doctor for a transvaginal ultrasound to monitor its size.
Most women get cysts. Your body usually reabsorbs the blood in time. That is why it sometimes takes months for cysts to go away. If your doctor who found the cyst didn't recommend follow-up, then chances are that your cyst is not worrisome for cancer. Most cysts do go away on their own. If you continue to have pain or other symptoms, it is always a good idea to check back with your doctor so he can check on your cyst. Best wishes.