Welcome Viktor,
How squishy is the plant? I see maybe a couple of spots that look like some kind of rot might be developing, but if the base of that plant is solid then most of what you are seeing would appear to be just normal subsurface coloration of the plant. So if the plant is solid, I would just leave it out to dry for at least another 3-4 days (depending on where you are make sure it is warm, but not in any kind of full sun), then plant in whatever medium you were planning and then just give it another week before watering a little.
If the plant is squishy, it probably means there is a lot of rot going on, then I would be inclined to take it back to the garden center and either get another one or get your money back. You might save it by drying it out, just keeping it warm and dry and out of the sun till it is not squishy anymore - but there is a good likelihood that is will just keep rotting.
As with most barrel cacti, re-rooting from a cutting is not a high success rate approach, not impossible, but highly likely to fail. I have heard of a success story here or there, but mostly have heard and experienced spectacular failures.
Good luck!
