This is nature's gift to you after the plant lost its growth point. Put a cactus on the brink of death and sometimes it comes roaring back as if to say I'm still alive! Stefan has summarized the situation nicely.
You might need to protect the new growth from hot summer sun. Try to avoid repeating history there as you might not be so lucky next time.
Your ability to start new plants from cuttings is partly dictated by their starting size, partly dictated by the season. So maybe in the fall when it's not so hot, or maybe early next spring to gain a size advantage. But not during the extreme heat of summer or the colder days of winter.
Rootless cuttings will benefit from some sun protection (ideally shade cloth or morning sun only, no overhead sun during the brighter/hotter months). Only after they are properly rooted and growing (maybe 6 months or a year after being started) can you safely consider giving them a more exposed location.
We can give detailed instructions on starting this cactus from a cutting, but I think it is best to wait before doing so.
