I doubt the papaya caused your period to come or we'd have a bunch of women starting their cycles every time they ate papaya. It was probably about to start.
I too have never heard of a natural way to deal with PCOS. Hormones seem about the only way, which is what birth control is. There are the options Anonymouse mentioned as well as things like the Mirena IUD, which has hormones in it, etc. You can ask if it's simply being tired of having to take a pill daily. Your gynecologist or physician may know natural remedies, too, but you'll have to ask.
I'm not sure there are any good natural methods for dealing with pcos. If it were just irregularities, you could just live with them. If you've got other symptoms causing issues, the hormonal methods are probably the best way to control them. What is it you don't like about the birth control pills? Is it taking them every day, or are you getting other side effects? If it's the schedule, ask your doctor if the nuvaring or the patch would have the right doses, since the nuvaring is once a month and the patch is once a week instead of a pill every day.
I love having birth control. Before I started taking it 5 years ago, I had 7 day long painful, heavy periods and several days of horrible PMS where I'd be an emotional wreck, crying at almost everything sometimes. I've been on a bunch of different ones over the years (mostly just what the teen clinic was handing out), and it's been a major improvement in my quality of life. I've found that if I'm not depending on them for preventing pregnancy, I can afford to be a bit lax in my schedule, taking them at some point in the evening and occasionally forgetting one, without having any spotting or other side effects. If you use condoms anyway and you're tired of the rigidity of the dosing schedule, you might find you could "get away" with being a bit more relaxed and still have symptom control.
If you're on birth control, you *shouldn't* be ovulating (although you still can). Periods on the pill are from withdrawal from the hormones. If the papaya did anything (and just because you got your period after eating it doesn't necessarily mean it did anything), it would be to make you shed the lining, and nothing to do with the follicle. The follicle ruptures mid-cycle, not during menstruation.