Cysts can and do cause pain. When I have had cysts, I have noticed the pain more when I was sitting down. But, you describe the pain in the lower right pit of your stomach. Is the area of pain higher than belly button level or lower? Pain that is higher is suspect for gallbladder issues. Every woman experiences ovarian cyst pain a bit differently. But, it is usually described as an ache, pressure, or sharp pain in the area that is about as low as the hip bone.
It is very important you get it checked by a doctor to determine what is causing your pain. In addition to your gallbladder or a cyst, your appendix is also on the right side. It sometimes causes pain for a period of time when it is inflamed. If it ruptures, it is a serious condition. So, getting it checked soon is important. I hope you find some answers soon!
I've had pain when sitting from ovarian cysts, not to mention bending, walking, lying down etc etc. I avoid hard seats and use a foam wedge cushion from a back care shop. I also find anything tight around the waist (jeans, tight skirts, belts etc) will make things a whole lot more uncomfortable when sitting so try wearing clothing that is loose around the waist -- I was surprised at the difference it made.