Mammillaria melanocentra is correct, I'd say.
You need to immediately take it out of the pot and check to see if it was potted properly, usually they aren't. Nurseries raise cactus seedlings in a peaty mix which is kept evenly damp, this works great for producing fast growth in a hothouse environment. Not good for normal growing, though - if the stuff gets soaked, it stays wet too long, if it get dried out it resists re-wetting. That peat should all be removed from the roots when they're potted up for sale, but often it's not. See:
http://www.cactiguide.com/forum/viewtop ... =25&t=9222
What's visible of the potting mix here looks good - a lot of pumice or perlite, not much organic matter - but if the roots are wadded up in a gob of peat you're going to have problems unless you fix that.
Once you see what the roots look like you'll be better able to judge whether to keep it in this size pot or go a little larger. the pot should be just a little larger than the existing root ball.
That watering advice is okay as a general guideline, but it varies depending on conditions - rather than water strictly on a schedule, you need to judge how dry the potting mix is. Generally, they need more water in hotter weather - but in extreme heat, especially if it stays very warm at night, they may go into heat dormancy and stop taking up water. In that case, you need to hold off on giving any more water or you could cause root rot.
In the winter, they should get a dormant period when they get little or no water. Ideally, they should be kept cool then (except for tropical species) - they can go without any water if it's cool enough, say 40 to 45f. Most people can't provide a cool enough place, though, which means a little dribble of water needs to be given periodically to avoid major root loss.You don't want to really soak the soil then, just give a couple teaspoonsful.
