What parts are dated and on par with the U.S. ? The U.S. has air superiority over Russia. I have limited knowledge on Naval weaponry so I can't really comment on that end.
Russian avionics are not believed to be on the same level as existing western platforms like the F-22 and F-35. The situational awareness that these aircraft have is remarkable and is a far greater advance that what you see on the skin.
The same can be said of much of the Naval sensor suites as well, don't forget the Russian Navy was largely left to rot for 20 years and is only now starting to rebuild and re-arm. They don't have the equivalent tech of the US Aegis and British Type-45.
That said, the Russians did keep their anti-ship missile programs well run. Much like the Chinese realise now, you can't fight the US Navy head-to-head, but if you can threaten a Nimitz class then you've got a good tool.
The P-800 (SS-N-26) is a good example of this. Fairly recently in service it's a capable missile. That's not to say it's a better system than the US Harpoon, or that a single missile could defeat a Super Hornet w/ AMRAAM, Aegis Missile shield or a CIWS defence. But you put enough of them in the air and you can create havoc for even the most organised fleet.
This wouldn't be unexpected for the USN. It's gone now, but the F-14 was basically built to handle this. The Super Hornet is short in some areas compared to the F-14, but it has a few advantages as well along with the fleet it's protecting being a bit more modern than it was when the 14 was around.
The Super Hornet would spank the F-14, with newer versions of the AMRAAM having similar range to the Pheonix.
And like I said above, a single missile would be taken out by any 3 lines of defence, but a multiple launch is a likely tactic.
It's not out of the question that the US loses a ship, but who ever takes that ship will probably suffer greater losses in the process. Still, that may be enough.
I think there's a huge difference between the US losing a frigate and an aircraft carrier. The form would be embarrassing. The latter would be a humiliation and would swing public opinion.
That could go either way, wiping out those responsible, or retreating with tails tucked between their legs. There's examples of both in US history.
Not sure how much that matters, US public opinion is running pretty strongly against this already. Besides, it's not like the US government cares much about what the public thinks, anyway.
Afghanistan and Iraq were fuelled on the anger of 9/11. Vietnam was fuelled by the fight against communism. I see a potential conflict in Syria following the latter more, but that's an outsiders view looking in.