How do I divide an overgrown cactus (looks similar to Mammillaria vetula subsp. gracilis Preiff)?
Very carefully so you don't get stuck. I have some kitchen tongs I use for things like that. I work the tongs in between two of the off shoots and then if they don't come apart from that I take a sharp knife and cut off an off shoot.
Then let it dry overnight and pot up in fresh cactus soil, something that is gritty and sandy, that drains well. I have re-potted lots of cacti.
I grew dozens of these things when I was young. The 'branches' will readily snap or twist off and will root if simply placed with the end in compost and sparsely watered. They then grow identically to the parent plant, and sprout their own branches, and the branches grow branches, and so on.