How do I save my cactus with damaged/non-existent roots?

That's actually just a cutting that never developed roots.  Put it back in the pot, let it dry out between waterings and give it lots of bright, indirect light.

Don't let it sit in really wet soil all the time or it will rot.

Stick it into moist sandysoil, but not too wet. With luck it will grow more roots. You can buy root starter to coat it with first if you want. Cactus plants are very hardy, so stick it in and give it time. The biggest mistake is to make it too wet. They like it with just a bit of water.