Pole Position won't turn on. Gee, I thought that was their natural state...
Welcome to the forums. And sorry that your first project is Pole Position.
The monitor is probably the easier fix. If it's making a constant ticking noise, then my bet would be that the horizontal output transistor is shorted. Since it's a newer monitor, it has a switchmode supply (rather than rectified line voltage), so the power supply tries to start, is overloaded by the short, and shuts down. This as opposed to blowing fuses like an older monitor. Find and remove the horizontal output transistor. Check it for shorts with your meter.
As for the game itself... May the arcade gods have mercy on your boards. If you've never done board repair before, Pole Position is *not* the place to start. They barely worked when they were new. But, start by checking voltages at the test points. Pole Position is notorious for poor power supplies and poor power distribution. Last Pole Position I worked on had a voltage drop of .8 volts from one end of the board to the other!
Also, check the edge connectors on the boards. They're well known for burning up. If they look dirty, clean them with alcohol. If they're burnt, then... you have more repairing to do, as the connector on the harness is probably melted. Check that too.
-Ian