Definitely a very poorly grown Crassula ovata. Personally, it looks like either a Gollum or Hobbit (two very similar looking varietal forms) to me. Fortunately, as w03 said, they are tough plants. Good odds it can recover.
When you unpot it, check the base of the stem where the roots are. If the stem is mushy, then cut off that portion to a healthy firm section. Let the cut callous over for a few days. (If the stem is good, then you can skip that step.) If the roots seem dead, yes you can cut them off. In either event, after potting the plant up keep it in a warm bright area but no direct afternoon sun. Bright indirect light should be fine. I'd suggest leaving the media barely moist until the plant has rooted.
If by "generic cactus mix" w03 means the crap-tacular stuff sold prebagged at the BBSs, I'm not a fan. Crassula are usually quite adaptable but a more porous/less "peaty" mix would likely be wiser -- depending upon your growing conditions. If your options are limited, you could try cutting the prebagged stuff with chicken grit and/or perlite.