There are scheduled flights between Kingston and MoBay - about $70 or a little less, I think. Air Jamaica and a small airline. Then drive to Negril. A flight to Negril would be very expensive for one person.

Remember to factor in time getting to the airport, although you could eat fresh fish in Port Royal and be at the airport quickly from there. Time getting your luggage in Mo Bay, getting your ride, and another hour+ to Negril. I would consider the bus as a very good alternative, the cheapest you will be able to do it.

I've only heard good things about the Knutsford Express, if that is who is doing the South Coast route.
I would encourage the Kingston trip - just as you have it planned. You'll be just fine. I spent quite a bit of time over 4 years as a solo woman in Kingston when I was working on a Medical project. I had a trusted driver to help watch over me, but I never felt unsafe.