Yes, you will want to have gritty soil on hand when you take action. If you can't get the cactus mix then regular potting soil mixed with an equal volume of grit (or more, if it seems rich) should work fine.
Timing is kind of important here, especially if this is an indoor plant and you are in the temperate northlands. There is no great need to take action right away, and spring is right around the corner, so why not wait for stronger light, longer days, and warmer temps before you repot, if that's where you live. Soon is good but right away maybe not.
Whenever you do repot, wait a few days to a week to water afterwards, especially if you handled or damaged the roots in the process. This is very important.
Two comments related to the plant.
There is a reasonable amount of diversity within Haworthia (and its siamese twin genus, Haworthiopsis). Some of these plants make really insubstantial roots, others make lots of roots, depending on the species. So you might want to choose different pot dimensions for your plant according to its needs. Seeing what the top looks like, as Daisy has requested, would allow us to give you more specific advice about this.
Whenever fat roots pop out of the holes at the bottom of a pot and start circling like that, you have to be really patient and careful when you get the plant out of the pot, trying not to break any of them if you can avoid it. Just watch the bottom when you take that step.
And since this is your first plant, especially given it's a succulent, I would emphasize the importance of strong light. If this will be an indoor plant it should "see" the sun for hours a day year round, weather permitting. That usually means a position right in front of your sunniest southerly-facing window (assuming you're in the northern hemisphere).
