I'm assume you mean composite or component, and RCA stereo audio output. If so, Ouya has only stated HDMI.
As the above said, you would need an digital to analog converter. An amplifier wouldn't help you in this situation. Unfortunately, you'll spend more on the converter than you would the Ouya. Well, maybe you can find an old used FuryHD for $100 to handle video. There are other issues you have to take into account such as will the converter tweak the signal to make sure your TV will accept it is a big one.
I had to go that route when I setup a home theater PC, and my HDTV predated DVI; it only had component HD input, but the HTPC had display port. I got an HDFury2, which just came out, and ran about $250 at the time. That was cheaper than buying a new good quality 55" HDTV.
Hopefully that was helpful.