Use an ouya as your platform. Side load it.
Use an android on a stick; side load it.
I have a PC in a cab with an original arcade monitor already... that's the platform I want to run this on... OUYA/Android platforms wont work for my needs (they couldn't run most of my games and even if they did I need VGA output for the arcade monitor, I'm not interested in adding lag to convert from HDMI) Becides, getting the software to RUN isn't the question I'm really asking here...
The core of my question is how to do you use one of the many android emulators within the context of an arcade machine (launching from a front end, qutting back to a front end, mapping controls etc)... I'd still have all of those problems using an OUYA or an Android stick.
There's an online version of the game ( http://www.flappybird.com/play/ ) that uses the mouse clicks in a web page so what you are looking to do should be possible.
I'm sure it's possible... There are a few different android emulators available the best of which looks to be Bluestacks:
http://www.bluestacks.com/