worst case you can work backwards to figure out the pinout if you've got a multimeter and a scope and some patience.
thankfully it looks like the primary function of most of the connectors is spelled out already.
start with writing down a list of connectors and a numbered list of each of the pins on each connector (there is usually a mark or some indication which pin is pin 1)
then pick a common chip on the PCB such as anything with a 74xxxxx. google the pinout and determine the 5V and Ground pins, then go through all the connectors and determine which pins have continuity. to 5V and ground and mark those on your list.
then you can start narrowing things down, look at the power connector and determine which pins ARENT power or ground and follow the traces to a chip, then look up the pinout for that chip to see what kind of power input that pin typically accepts.
for the video connector you should expect a ground pin, a sync pin (Which will have a 15kHz pulse) and 3 color pins which will have floating voltages you can use a scope to determine the pulse pin and the color pins.
try to identify an audio amplifier chip, look up the pinout for that and follow the output traces to determine the audio output pins. you can also poke around with a scope to help with this.
then that leaves the input pins we know a trackball typically has a 5V, a Ground and then 2 plus pins. once you've identified the 5V and ground pins on the input connector and you've got the audio and video working to a point where the game starts up and you can hear and see it, then you can take a wire and connect ground to each of the unknown input pins to see if you can figure out what they do.
