Ahh, I thought you were referring to the last demo scenario.
I too played this as Russians versus AI US, and got a decent Russian victory. As
@JTimo indicates, I recall focusing on keeping the US at arms length (i.e. outside the walls), and ensuring I was selective in exposing my troops positions through keyholing, so they could interdict US troops in the open, without being exposed to too much US support weapon/team overwatch.
Also, split your squads, all of them. Munitions are super-lethal in this modern era, and will easily destroy an entire team if they are caught clumped together. And the opposite is also true: your infantry firepower is so much more powerful than that of WW2, so no need to rely on a whole squad to suppress the enemy: a 3 man team is is quite potent. Spread those teams out over different floors/buildings to get overall overlapping coverage, but with each team individually having limited LOS angles/exposure to the enemy.
However, the BIG differentiator between playing against the AI versus against a human will be the AI's inability to use arty and area fire. A human BLUFOR player will suppress the hell out of all likely Russky locations, and pound the more likely exposed ones with arty. AI can't do any of that: the AI arty observer (I think there is one) is useless.