I think that the installation of a game only makes it less noisy and easier on the disc drive. But the RROD problem is still there because of the 90 nm processor. So unfortunately it can still red ring - hopefully you have a warranty. If not I think it'll be about $100 to fix?
I originally had an Arcade with the new CPU, but I sold that a year ago when I needed some money. I was desperate to get a 360 a few months later so I paid $120 at Gamestop for an older Pro model, and I'm not sure if it has a Xenon CPU or not. It's been about 6 months and it hasn't red ringed. How can you check?