My first workings on my cab used a PC monitor, front mounted buttons like youve got, including power. With the bezel in place there was maybe 1/4" at most between the button and the bezel. I found some round adhesive velcro dots and attached em to the 'inside' of the plexi bezel so that they lined up with the respective buttons on the monitor. A little push on the plexi at the right spot and it would cause the velcro dot to connect with the button and do its thing.
Depending on the space, you may need to use like an eraser nub, or a piece of dowel rod, etc so that at rest, the button pusher is close to the button
Alternative would be to make a pinhole thru the plexi, lined up with the button, and use a toothpick or other pokey-thing to activate the button
Now though that Im using a TV, Ive got a remote. Also the tv does auto-power-on so I rarely even need the remote
If youve got a big gap between the front of the button and the back of the plexi, you could try a leaf-switch button assembly(just the round button bezel and plunger assembly) This might be cool for a power button for the monitor