A general physician and probably dermatologist can do it in their office, but not sure on cost. Depending on your insurance it can be just the cost of the co-pay.
I had a cyst removed from my upper back earlier this year that was inflamed. It's important to get all of it out, otherwise you risk having it come back... and properly following up with packing and dressing the wound (multiple times, depending on how deep the cyst was) is just as important. Even with a few shots of lidocaine, it hurt, and after it wore off I was in agony until I got some Tramadol.
If your cyst is intact, not painful to the touch, and not red, it's probably OK to leave it until you can get it properly removed. But if it manages to rupture you'll need a doctor to take care of it ASAP. I had mine for many years until one day while snowboarding I fell on my back and I think it ruptured a little - it was painful when pushing on it and constantly red.