Been there a couple of times. While it will take some time it's not really all that hard. A schematic, meter and patience will be a big help.
It looks like you'll need connectors (either EM style jones connectors or more modern Molex connectors), extensions for each wire and heat shrink tubing. You might start by picking a type and location for the connector and then extending each wire to reach the connector with some slack to spare. I'd twist, solder and heat shrink the extensions to the wires although crimp style butt connectors would work if done properly. Then pick out, verify (with the schematic and meter) and install the obvious wire color pairs into matching parts of the connectors.
To verify a match, jumper the loose ends together then ohm out a point in the cabinet against a point in the head. For example check that the resistance between a 10 point target on the playfield and a solder lug on the 10 point relay in the head is an ohm or less. Once you're sure you've got the right ends jumpered together install them in the connectors. Then reverify the same two end points to make sure that the connection through the connector is good.
If you're lucky you'll be able to visually match up most of the wires and what will remain will be a smaller number of faded wires to sort out. Those may take a couple of guesses to find the match but each match will make the remaining matches easier to find.
Don't commit anything to the connector without first verifying that the appropriate end points are electrically connected because debugging a problem created by a mismatched pair is much harder than verifying that the matched pair is correct.