How could someone get peritonitis from an ovarian cyst rupture?

Any irritation : Of the peritoneal lining of the abdominal cavity is peritonitis. An ovarian cyst can bleed when it ruptures, and blood can cause a local peritonitis. But this is way different from bacterial peritonitis, which is what results from perforated GI organ, like appendix. These are much worse, and can actually be fatal. So seriousness depends on the source and type of peritonitis.

Yes.: It would be uncommon but possible to get peritonitis from ovarian cyst rupture. This is particularly when there is significant amount of internal bleeding which may irritate the peritoneum.