depends how they're mounted, a lot of cheaper joysticks can be "all in one" where the cable is kinda entombed inside the button at manufacturing stage. You can still cut the cable that goes to the buttons, though, but you'd either have to solder them to the microswitches on the buttons on your cp or use cable clips to attach to the cable then to the microswitches on your cp. If you know NOTHING about soldering then I would suggest going with this:
http://www.ebay.co.uk/itm/Zero-Delay-USB-Encoder-PC-to-Joystick-2pin-Happ-Push-Button-For-Arcade-MAME-/191558261745?As you can attach everything from your cp to that (if you have LOADS of buttons you might need 2, check the listing for how many it supports). You can see from the pictures that all the wires are intact and ready to plug and play. I've used a couple of these recently and it does certainly save a lot of time compared to hacking joypads. Unless you're really good at soldering and have a secure mounting inside the machine for the hacked pad it's also a lot more durable (badly soldered joints can and do pop off from time to time if you've made a pigsty of things)
You can use this then use the program I mentioned above to make the pc think the data it's receiving is from a 360 pad.
I'm presuming you're not wanting an analogue joystick on your machine, right ? (being as the 360 pads have 2 analogue sticks I'm just wondering. That's harder to implement)
I pretty much gave up on trying to hack 360 pads for arcade boards etc and just used other controllers. I was buying repro snes pads (with usb connectors, not console connectors) from ebay for a while as they only cost like £1.50 (or did at the time) with free postage :-o Can't say fairer than that and you get a load of terminals to solder to, 4 way d-pad and 8 buttons is quite good for that price BUT of course you do have to know what you're doing with soldering. I never had an issue making these pads work ingame as anything I tried accepted them as default controllers and, of course, if you could remap ingame it was even more helpful. As with the software I posted above, there's a lot of external remapping programs for games these days that override the games settings (or inability to recognise your controller). Pinnacle game profiler is another one, but I believe you have to pay for that. there's also joy2key if you can remap the keys ingame then that can help, too.