How can I airplay to my Apple TV without having to turn it on with the remote first?
The short answer is, that you are doing the right thing, the Apple TV should just wake up when it gets an AirPlay stream coming at it.

My Apple TV (720p) one is having similar issues, you have to wake it up to stream to it, and it has other issues.. I would suggest a factory restore which is what I am going to do.

It will loose all your preferences, but should fix the issue. Worst case you could also look at reinstalling the firmware using a Micro USB cable and iTunes on your PC or Mac.

Instructions on Apple.com for both methods