I'm not familiar with this board so we're going to just do some basic diagnostics to help rule out some issues.
You can use these techniques for simple board issues, they cover some common stuff.
Audio
Let's check if the amplifier is working. It's easy!
Turn the volume potentiometer to the middle point and turn on the board.
Now run your fingers under the amplifier and surrounding capacitors. Got some crackling/buzzing sound? If so, your amplifier is good.
No? Let's rule out the potentiometer. Set your multimeter to resistance mode and at least 2 kiloohm range. Check the resistance between the middle pin and the other two. Move the pot to different positions and verify that the resistance changes. If it all checks out, the pot is good.
Controls
Let's see where the jump button goes and make sure the board is receiving the input.
From the JAMMA edge connector follow the traces that go from the button inputs to the board. You'll arrive at one of the resistor arrays right next to the board connector, marked RAxx, xx being the reference. The resistor arrays will have markings. The number is the resistance between pin 1 and each of the other pins. E.g., if it's 4.7 it should be 4.7 kiloohm. Turn the board around and measure the resistance between pin 1 and each of the others. They all should be around the stated value. If any are grossly different the resistor array needs to be replaced. It acts as a pull up on the input signals so when you press a button the signal actually goes low.
If the resistor array is ok, follow the trace to the NEC ICs nearby.
These are optocouplers. They isolate the board from voltage spikes coming from the inputs.
You'll need help for this bit as you'll be using a multimeter and pressing buttons at the same time.
Set the multimeter to voltage mode at a value over 5v. Place the black probe on ground (on the connector for instance).
Place the red probe on the pin where the input goes. Measure the voltage there, it should be around 5v.
Press and hold the button. The voltage should be zero.
One of the pins on the opposite side of the one you've identified as where the jump input goes should toggle as you press the button.
Check the datasheet to see which one.
Repeat the process for the pin(s) on the opposite side. If they go from 5v to 0v, they are ok. If not, replace the optocoupler.
That's it for now. A logic probe helps immensely for this type of thing, try to get one when you can.
Good luck!