Not sure what type of monitor you are using, but have you experimented with the monitor yet to see if it will turn on with power? Many of them will automatically power up when the power is added. Some have a setting to turn that feature on. If that doesn't work, Another common trick is to jam the power button down permanently, and when powered up, the button will trigger and turn the monitor on. Try holding down the power button and plug it in to see if it turns on. If it works, you can shove something in the power button hole to jam it down.
If that fails, a more dramatic method is get a so pop out power button, solder wires from the metal leads underneath to a switch located elsewhere on your cabinet. To make it one button startup, use a smart power strip and have the monitor plugged in as the "master" Once the monitor is powered on, everything else on that power strip gets power. If the monitor is turned off, the power cuts to everything else. suddenly the monitor becomes the main power button.
If you want a less invasive solution to the monitor, you can drill a hole through the wood where the monitor power button goes, then create a rod that goes through that hole and acts as a pushbutton. A power button on your monitor area might be annoying, but I have seen it done.