Even if the N1 worked it wasn't likely the Soviets would have used it to be 'second' to the Moon as the mission plan was quite obviously and well understood to be much lower quality than the American Apollo program. What they really needed (assuming they have the will to actually admit to still competing with the Americans*) was to be "second" to get the there but do it BETTER than the Americans. (While telling the world "We weren't "racing" the Americans, we were just planning on doing it RIGHT instead of first" )
Well the thing is the Soviets were actually working on a Lunar lander that was better than the Apollo mission, it was called the L3M and the 1972 variant of it would have accomplished the goal of being better than the Americans with bringing 3 men onto the moon and would've been able to stay on the moon for 90 days.
The moon lander would have an enlarged Soyuz descent capsule cocooned instead of leaving it in lunar orbit and 23 metric tons in total would be landed on the lunar surface.
Two N-1 rockets would be still needed to launch and land the L3M on the moon with one carrying the crewed lander and the other carrying the crasher stage Engine block that would needed to propel the L3M onto the lunar surface safely.
The L3M itself would travel back to earth and burn up with the descent capsule being hatched out of the cocoon and safely returning to earth.
Cutaway view of the L3M-1972
The Soviets would've likely have buried and hidden the true reason of the L3 Lunar lander to the world and declare that they were totally not racing the Americans at all and were instead just working on a lunar lander that better than the Americans when the time came to publicly unveil the lander.
This plan will likely never happen and probably would've been killed by the leadership if it was ever proposed to them.Lets say that using the Proton/R7/Soyuz/Salyut technology the Soviets put a modified Salyut/Almaz into orbit along with a 'transfer-stage' capable of getting it to Lunar orbit and back. Attach a couple of their Lunar landers and send it off to the Moon where they land only two men but they do so in two separate locations that Apollo never visited. Or better yet they make the landers capable of multiple trips and they land the two men several different places each and bring the whole thing back to Earth orbit implying they can do this as often as they want. How does the US respond?
If I were to think on a late 1970s/early 1980s response with the Space Shuttle the only thing that comes to mind is the Shuttle-C type proposal to bring Americans back to the moon as the Saturn rockets are well beyond saving by the 1970s. Unfortunately whatever response the Americans have is likely to get doomed by budget cuts from Congress by the time the Eastern bloc collapses and the USSR soon after kicks the bucket.
The Americans might just stick to the futuristic Space Shuttle as their main response to the old N-1 rocket in the end with maybe a working Shuttle-C rocket and a Space station in orbit by the time the Soviets fall in the early 1990s. But that's the best that they will probably achieve.