I am a RN and have seen folks get this surgery tons of times. It is almost always laproscopic (2-3 very small incisions) and you are back to normal within a week or two. Often the final decision on whether to just remove the cyst or remove the ovary will be made by the surgeon when he sees what is going on. And that may not even be an issue depending on the size seen on ultrasound. In the grand scheme, it is not really an issue since you have 2 of them and fertility remain intact with one gone.
My ex had a cyst removed and wished she had the ovary removed because the cysts returned. So that is something to keep in mind.
Best advice I can give you is keep a positive outlook and get up and moving ASAP after the surgery (Do not run a marathon, but do not firmly plant on the couch either). It promotes healing and prevents complications.