Ah, I missed that ("which does not have DVI inputs" right in front of me) - DVI/HDMI is clearly out of the equation. The DVI connection (and HDMI , since it borrowed DVI's video standard) is purely a digital video connection, and your display's RGB inputs are analog. There's no way to use the AppleTV's HDMI output with your display.

Since you don't have a lot of video sources, you may want to migrate over to using the 990's video switching. Connect the VCR (composite), DVD (component), and AppleTV (component) video outputs to inputs on the 990, then connect the 990's component monitor output to the Sony XBR and leave the Sony on that video input. The 990 can then provide video switching at the same time that it switches audio. It will even transcode the VCR's composite signal to component.