It's usually just a matter of following the traces from the joystick and button contacts. All the buttons should have 2 contacts, which are closed by a switch of some kind. One contact is positive and one is the ground. Just connect one contact to the NO on your arcade buttons/sticks and connect the other to COM. Do this for each set of contacts.
The easiest way to solder wires to circuit boards is to drill a small hole (about the size of your wire) in the end of one of the copper traces (where it goes to the button contacts) and scrape away any green insulation from that area. Then pop the end of your wire (preferably solid core wire) thru the hole and hold your soldering iron so the tip touches both the wire and the circuit. After a second or two, start feeding solder into the joint, until you have a nice little bead of solder. Withdraw the iron and hold the wire still for a few seconds while the solder cools. That's it!
Good luck.