I had a cactus that looked similar to the picture of yours. I never
knew its name. Mine (also inherited in poor condition like yours)
broke off one day when I was moving it. I kept the top section and
let it air dry for several days to seal off the "wound." Then I just
put it in some cactus planting media and watered it occasionally,
letting it dry out well between waterings. I also put a stake in the
pot to hold it upright until it established a new root system. It did
survived, although it never was particularly attractive or interesting
to me. I finally gave it away. I'm not sure that yours is the same
type of cactus as what I had, so you might wait for other responses
before taking any drastic measures.
I'll be curious to see what others have to say about this.
Good luck,
Gary