I am actually not kidding.
Is there a way that we could end up with a Russia that is far more eastern influenced, to the point that it is Buddhist, but still obviously a Russian ruled state?
Is it possible?
Mongol Golden Horde in 1200-1300's converts laregly to Buddhism instead of Islam. Ozbeg Khan doesn't take power in the group, and his place in OTL is taken by one of the rival Buddhist Mongol princes.
And to do that, you'd have to have much more intensive links between the Buddhist areas of the Mongol Empire, and the ulus of Jochi...literally thousands of Buddhist clerks and clergy, for example.
Not impossible, but unlikely given the antagonism between Jochi's descendants and those of Ogedei.
I put the pre-Muslim Golden Horde out there because their is no other time and no other situation in Russian history that would allow for an influx of Buddhist ideas and traditions. There were the Kalmyk migrations into the Pontic Steppe, but they were weren't nearly as powerful as the Mongols.
...and by then "Russia" was very orthodox, to boot. Even if the Tsar managed to work out a deal with the Khan and the Emperor to have th Dzungars more to Russia en masee, it wouldn't have produced the desired outcome.
No, since Buddhism would be critical to his wars...It seems difficult for anything to displace Christianity from Russia, particularly something as geographically distant as Buddhism.
So what if we consider a situation where Russia was never Christian in the first place, perhaps because Christianity itself never arose?
If the Roman Empire doesn't come under such a proselytizing religion, then Russia could perhaps be open to Buddhist steppe nomad conquerors (who may or may not be the Mongols, given butterflies from the early POD).
Another thought: could Buddhism spread to the West under the patronage of Alexander the Great? Would he want to do that?