Your "Old Lady" cactus is a Mammillaria, probably Mammillaria hahniana. Mammillaria (and any cactus with white hair) is very susceptible to over-watering. I doubt you can save your plant.
For next time: The pot its in is way too large - there should be only about 1/2 inch of space around the edges. The pot your cactus came in was probably the right size. I doubt you would need to repot it for at least a couple years.