I can't think of any reason why that wouldn't work. I used to have to do something similar for a different device, when I didn't have a TV with HDMI inputs.
Here is one for $4.88 though. I always buy my cables from eBay. They tend to be cheaper 99% of the time.
I bought an HDMI switch from Monoprice.com. I do not want to post the link, because I have already had posts removed, as it is not permitted in these forums.
A switch may or may not work for you, as it is hit-and-miss at times.
I got a switch that had RCA component out as well. My TV only had a DVI inputs. I assume that your TV does not have an HDMI input, as if it did, it would not have a need for RCA component audio in.
This switch allowed me to route everything from the ATV2 into my TV with one device. The switch was about $60 USD and the cable about $7 USD. That was a small price to pay, considering the cost of some new TV's.
Your TV needs to be capable of 720P and be HDCP compliant though. Confirm that if you have not.
I hope you are able to get it up and working. The APTV2 is a great device, but has some compatibility issues, even with fully compatible new TV that have HDMI and are HDCP compliant.