It appears that your power button is a momentary button (you push the soft button on the panel and a material under the rubber button contacts the ring-shaped pad directly below it).
About the only way to see if it'll be easy to do is to see what happens when the power button is depressed permanently. Try this: Remove the battery, find some way to fix the power button down (don't let it "bounce", since the momentary off-on is what should normally make the radio "wake up"). Once the button is held down, insert the battery. If the radio turns on, then all you'll need to do is open the case and find something that you can fix to the button's contacts to make it stay "pushed down". These are typically capacitive effects, so you may have to experiment with a few items to find something that will work.
In a pinch, you could just stick something on the power button and tape it down, so it's always pressed down. It be kind ugly, but it could work.