You should see two stems come from the pruned top. As for the piece that came off, you can let it scab over and lay it on the soil. I hope someone else will advise you on rooting. People have their own preferences and I have not had enough success.

Btw, it is not a cactus, but it is a succulent.

Where is the "root-like thingy" growing? It sounds like air root that some succulents grow.

I know you didn't ask - but what kind of soil is it potted in? and is the pot sitting in a glass bowl filled with that colorful sand/glass?

The potting mix looks to be a combination of small pebbles and loose soil...but i’ve never planted anything before so i’m not sure. Yes the pot is sitting in another glass pot filled with the coloured sand. The root-like growth is very tiny, and there is only one strand:

(this was the closest i could take with my phone)

It's a Crassula of some kind, maybe perforata, rupestris or marnieriana.

Did they give you instructions on how to water this thing? Are you supposed to lift out the tiny inner pot to water?

As Karen suggested, you would need to lift potted plant when watering out of the sand, or excess water will just sit at the bottom and will, possibly, cause rot.

If "thingy" is a root and not a piece of torn flesh, plant is starting to grow roots where it was broken. So cut if off cleanly and completely (not the root, just the stem) and stick into well draining mix.

Original stem should grow a branch or few from broken top. Plant should be receiving good amount of light in order not to etiolate.