If you get any of the grocery food mailers you can use that, fold it and stash it somewhere for reuse. Make it long enough to go around the cactus with a bit left over that you hold like a handle. I've heard of people using tongs and cloth bandanas, but the tongs could damage the spines, and the spines would be getting stuck into the cloth, which again ends up damaging the spines with more work for you. Newspaper is the easiest, and you will no longer limit yourself to plastic pots for them.
Photo is from
BBC Gardeners World: