With a post-1900 PoD....
1) Reverse every ill that has befallen Russia in the last century (destructive invasions in both world wars, civil war, the rise of communism, gross economic mismanagement, the break-up of the USSR). Given that despite all these set-backs the country managed to still have a massive population (ahead of the U.S until the fall of the USSR), massive economy and a massive army, a non-russia screw world will see it as the definitive superpower that cannot be challenged by any of its European peers.
(Step is required to give Russia long-range deployment capacity to conduct such an operation on the other side of the world)
2) Screw over America as much as possible (even worse racial tensions that are ignored in an alliance between northern business and southern farmers, a progressive era that is less successful at "defusing" tensions in populations that wanted more liberty, more economic mismanagement and an equivalent to the great depression that is massively increased in magnitude) - leading to an outright break-up of the country that at the very minimum sees the industrialized north be held by a different government than whoever holds georgia.
(Step required because even a weaker "great power" focused on its navy can usually successfully defend itself stronger superpower if it has the same geographic advantages that America has (see UK vs Germany, then consider how much wider the Atlantic & Pacific are than the Channel).)
3) Create a "base" for Russia in North America (preferably continental America near the eastern seaboard). This can be a genuine ally or a client state, but extra points if it is a former part of the USA that is at odds with whoever holds Georgia.
(Even without a powerful hostile navy, an overseas deployment of this size needs nearby bases from which to be launched, can't just put people on a ship in St Petersburg and land them in Georgia a month later).
Alternatively, if you want something in a world closer to OTL....
1) Russian sub gets lost, crashes into something, makes landing in Georgian cost
2) Russian soldiers have lost radio contact with home (equipment maulfunction) and for some paranoid reason believe there is war going on
3) They raid some nearby town for supplies and barricade themselves
4) Siege and/or manhunt follows until they are all dead or captured.
5) The US and Russia say angry things to each other but ultimately do little.