A multi meter is good to test the required recommended voltages however being that you said your hooking this up to a 6 in 1 switcher PCBs sometimes certain boards run better at lets say 5.2v as apposed to 5v. If most of them and I'm going to assume they are JAMMA on that switcher should require -5, +12v etc. I wouldn't really worry about blowing anything up. Turn the power off on the machine, unplug the game from the wall as well. Simply as you said remove the older wires 1 at a time or label each wire, AC / Ground, +5, - 5, +12 etc.. and place them onto the correct corresponding terminals on the new switching power supply. Another thing to note is that sometimes new power supplies they are set to 220v on the power supply, sometimes 110v, obviously you want to make sure if in the USA you have it on 110v. probably the hardest part of installing a new power supply is not losing those tiny screws which attach to the mounting brackets that happ provides you with. Also if you find your not getting power after installing if your backdoor is off or it's possible that your interlock switch might not be engaged properly. If you want to test the game with the backdoor off simply PULL on the interlock switch, you don't need to use rubber bands or tape.
After removing the original wires from the older existing power supply It's also not a terrible idea to strip the wires down if they are corroded or show signs of rust etc.. If you have spade / quick disconnect type of connectors those are very simple to slip under the screw terminals. If your just using bare wire fully remove the screw from the new switching power supply and create a hook / J around the screw and re secure screw back in the screws. This should create a nice tight connection even if the game gets moved. Also if you don't have a way to label the wires (painters tape / masking tape and a pen / marker) and insist on removing them all at once a camera helps just in case you forgot how they were. Of course you can always simply trace the wires to the harness and look up the pinout but that's could become time consuming.