First be sure you've got a template so you know where you will be positioning your joystick compared to your buttons. I use one I've printed out on paper to mark my desired holes.
At this point, there's two ways to mount a joystick - from below and from above. Most people choose to mount from below since then no screws are exposed on the top of your CP. Also this means you simply have to drill a 1 1/8" hole (just like you would for buttons) for your stick to poke up through - no need to cut a big hole for the joystick assembly. While you're right that many people who bottom-mount use a router on the bottom of their CP to rout a rectangle in which the joystick plate can fit, the primary reason for this is to ensure the stick has as much length as possible sticking up through your CP. You can always do it later if you feel your stick is too short.
I haven't honestly used 1/2" MDF but it sounds worth a shot if you've already got some laying around. And if you're not using 3/4", you probably don't need to worry about routing a rectangle for the joystick to fit in from below -- just predrill and shoot some wood screws through the mounting holes from below (make sure they're not longer than 1/2"!
As for CP vs full arcade machine, doesn't make a difference - all the above would apply. Note that the first attempt at any Arcade controls I did was 3/4" countertop material and I didn't use a router on bottom mounted (magstiks) joysticks. Worked just fine though the sticks did feel a bit short poking up through.
my 2 cents
