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I've never installed ventilation in a cabinet, but I would drill a bunch of holes in a circular pattern similar to how it looks on the back of some computers. Drill a few more for the mounting holes and use screws to mount the fans. If you don't have a drill, you should be able to cut particle board with a utility knife.

put your fans at the top but as exhaust fans, i.e. they are sucking the hot air out of the cabinet. Mount with bolts and nuts with locking washers.

make some holes at the bottom of the rear panel, to draw in the cool air when the fans are running

as to how to cut, drill a hole then use a fine toothed jigsaw

I use a couple 120v computer style fans plugged into a 12v activated relay plug strip. When my audio receiver is turned on, it turns the fans on. Can also put them on a traditional resistive dimmer to slow them down for noise if needed.

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As others have said, exhaust rather than intake.

Use a 230MM exhaust fan to pull as much out as possible, and if you’re willing, add two small 50MM case fans to create intake at the bottom.

You could also stick a bowl of cold stones near a natural air intake to the cabinet during gaming sessions.

Additionally, if there’s a way to command the Xbox fans to run at a constant speed rather than ramping when needed, it will allow for a more continuous temp control.

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I’ve always been taught to install enclosure fans blowing into the cabinet with the idea that you’re slightly pressurizing it and therefore dust isn’t being drawn into it through the seams. Genuinely curious what the reason you’ve heard for the opposite?

Just using normal physics. You can't add cold, you can only remove heat. Exhaust fans work with this principle and not against it.

where were you taught this? to remove the heat you want to suck it out of there, not keep it in; and you want to draw in cool air at the same time, so you get a circular flow of cool in from the bottom, and warm out at the top, also heat rises so it's a natural flow assisted by the exhaust fans; dust will be blown in by the fans in your scheme, much more than any drawn in 'at the seams', which won't happen since air will be drawn in the holes drilled at the bottom.

if dust is a concern put some thin cloth over the intake holes and remember to vacuum them occasionally

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Adding cold air however is a breeze.

Both positive and negative pressure systems work, and both have their own advantages.

In OP's case i'd go for a balanced system: two low speed 200mm fans, one intake one exhaust, both with filters (to keep pressure balanced).

Personally I'd cut a hole in the backing exactly matching the XBX, and place it on its side blowing out that hole (so top facing backward). That solves two problems in one, you're ventilating the AV cabinet and removing the largest source of heat at the same time.

Having the bottom of the XBox facing forward might be a bit ugly, though, if you care about that. If so, you could also do the same but blowing in - A bit less efficient (since you're venting the heat into an enclosed space), but should still basically do the trick.

We do this at NREL for plc enclosures installed outdoors specifically. We always have a fan on one side and a louver on the opposite (both with filters), so the airflow is basically the same whether your fan is pointed in or out.

Either way you’re just getting airflow from the outside, so the cooling capacity is basically the same. The airflow patterns will be different inside the box of course, but either way you’re just exchanging air

Recently installed some of these. Went very smoothly

It's not enclosed, it's a cubby without a door, so right now it's venting out into the room but the intake through the back of the cubbie is currently restricted to the size of the hole the wires pass through.