What is dripping? Is there a hole in the roof over the pump? Can you post a photo? I can't figure out why an old well is dripping.
Is the well one of those with a hand pump? Sounds like it from your comment about pulling the rod. If it is an old hand pump, you should be able to find someone who does antique things to take it out for free.
It should be pretty easy to get the pump and rod out. Then you want to cap the well so little thingies don't fall in. If you keep it available it is a feature that you could advertise when you sell the house. I hate to see someone drop old rods and stuff into a well. Someone may come along in 20 years and want to use it.
If you are buying city water to water your flowers and lawn, you might want to use the water from the well for that. You would need some kind of electric pump.