That 12 port patch panel you linked is designed to be mounted in an enclosure. The picture shows what looks like the white D89 bracket behind it - but the description doesn't mention it. The patch panel is sometimes sold without the D89 bracket.
I am using that same patch panel design, but I bought the one with keystone inserts. If a connector goes bad, I can replace it. And, I have a mix of cat5e and cat6 - not a huge issue, I now realize.
That patch panel will stand off the wall a couple inches, supported by the D89 bracket.
I suggest you use a more attractive, flush mount enclosure, instead, if you really want to use that 12 port patch panel. Enclosures come in a variety of sizes, and are usually designed to fit between 2 studs. The cable bundle could exit the enclosure through a bottom knock-out (hole in the enclosure), and then pass through a scoop or grommet mounted in the drywall below it.
Make sure the stud bay doesn't have any light switches on the other side of the wall. Start with a small-ish hole, and look inside.
I'd choose an enclosure that doesn't say 'Open House,' or anything else on it. Some have a hinged and/or locked cover. You cut the drywall, drop it in, and screw it to the adjacent studs. Easy to hardwire the outlet, within the enclosure, if there is an existing outlet below. Enclosures may be steel or plastic, and can be surface-mounted on the drywall, or flush-mounted/recessed (much more attractive).
If you choose your enclosure wisely, you'll have enough room for a replacement switch, when your current switch craps out. The enclosure should have one or more rectangular knock-outs at the bottom, for an outlet or 2, to supply items in the enclosure (switch, modem, router). You could also mount an AC strip in the enclosure, if you need more outlets. You shouldn't see any cables, using a flush mount (recessed) enclosure.
Make sure there aren't any obstructions in the stud bay - light switch on other side of wall. Start with a small-ish hole that's easy to patch, to look for pipes.