What is the diameter of the casing and what material is it made of? It is not beyond the realm of possibility that it was washout bored. My father and I went down over a hundred feet using nothing more than a pump and human power.
I would be leery about the air quality at the bottom of the old dug well but the cap on the casing should be reachable with little risk. None the less, have a buddy topside and tie your self off with a rope just to be sure.
One-time shocking of the well probably won't get rid of all the bacteria creating the smell so you may need to have a feed system. Drilling a new well could yield the same water quality issues. Either way, you won't know until you try.