Airport express and airplay. Is there a way to auto turn on amplifier?

While in terms of the specificity of the desired implementation I agree with @patrix, you could get the desired solution if you're open to reframing how it all gets handled. I'd say the easiest (but somewhat costly) way to get the desired behavior is to get a Logitech Harmony remote.

If the amplifier pre-dates remote controls, you might need to think further outside the box and look into a wifi/BT-enabled power switch (plug the amplifier into said switch and have the hardware power toggle on the amp always set to "On"; give the signal to the power switch to let the power flow through it, the amp then remotely powers on).

Things get very interesting in these days of "Internet of Things"… :-)