Well, the Russian Air Force definitely doesn't have the same readiness that the US Air Force has but AFAIK they should still be able to keep several Il-78 tankers in the air which should be plenty to give their strategic bombers protection (officially they have something like 15 - 20 tankers and even considering the poor shape that parts of the Russian military is in at least half of those should be operational). As for the strategic bombers themselves, both the Tu-95 and the Tu-160 have an operational range in excess of 10,000 km without refueling. I suspect they'd still hurry up and ferry aircraft eastward (if they haven't already started doing so just in case, one of the reasons the conflict in Georgia in 2008 went the way it did was because the Russians had a considerable number of troops ready not far from the border).
Then there's the matter of ground-based AA sites, most of North Korea's AA is around Pyongyang, the DMZ and the area just north of the DMZ, what they've got in the north of the country is little more than a token force and the combat readiness of a their equipment seems to be a lot worse than it is for Russia. There's also the fuel and ammunition issue that I mentioned, waging a long-range air war would definitely be costly for the Russians but they could do it, the North Koreans on the other would quickly run out of fuel with small chances of getting more. Combine this with the fact that a large part of the North Korean fleet is made up of planes so old they are just barely capable of carrying air-to-air missiles and I just find it hard to believe they'd stand much of a chance of doing serious damage before Russia strikes back. Admittedly North Korea does have a few S-300-derived SAM launchers but Russia has close to a thousand S-300 launchers as well as a few S-400-equipped battalions, including one stationed near Vladivostok.
I don't doubt that in terms of lost aircraft and soldiers a war with North Korea could easily be more costly for Russia than for the US but I suspect if it did happen the Russians could still do a lot of damage quickly. There's also the political side of things, if it did happen Russia would probably like to occupy a decent-sized chunk of North Korea before the international community intervenes in any way to avoid having a US ally right on their doorstep (I doubt the international community would side with North Korea but from Russia's perspective it'd be better to at least already have their own "peacekeepers" controlling North Hamgyong rather than UN/US forces within spitting distance of the current border).
Of course, I'm just speculating but I just don't see North Korea standing a chance against any of their neighbors in a all-out conventional conflict, were it not for the ROK being a US ally that would probably be their best shot, Russia and China are both capable of hitting them way too hard if North Korea tries anything though I doubt either one really wants to get involved in a war with North Korea precisely because it would be the kind of war that's less about precision strikes and more about taking out large amount of troops, equipment and infrastructure. To be honest I think the risk of a war on the Korean peninsula turning into another Iraq or Vietnam is slim, they have no strong allies like Vietnam and culturally and politically the situation in Iraq was completely different (North Korea's neighbors are a lot more stable than the Iraq and any power vacuum in a conquered North Korea could be dealt with more like Germany post-WWII, a de-juchefication similar to the sometimes overly pragmatic de-nazification of Germany would probably be the way to go).