We seem to agree on the fact that stealth does not mean invincibility. And it has to be said that there is indeed only one confirmed downing of an F-117 (and possibily another one hit but not not fataly damaged). But I have to point out it was downed by a single ‘lucky’ SA-3 launcher, not nearly the integrated air defence system it was designed to counter. Also succes during Desert Storm against Iraqi air defence by account of massively overwhelming numbers and technological/training advantages. I would not use those results as a gaurantee you’ll be able to suppress or overcome modern Russian integrated air defences.
As far as I’m aware the F-22 does not have an EOS system on board, so the only way to stay invisible in my opinion (no radar lock-up, which would give the Ruskies advanced notice and a rough position) is to creep up on unsuspecting fighters hoping to get in Sidewinder range before they spot you. They will have to rely heavily on AWACS-support to gain and maintain situational awareness, so guess what every single Russian pilot and missileer will be hunting for. Once the AWACS-support is dealt with, you’ll have to rely either on ground based radar, or the F-22 has to turn on their radar, which again negates the stealth advantage.
I have no doubt the US airforce will turn out to be the victor, but it will not be a walk in the park.