As Peter has already said, the border is currently open and there are thousands of Russians trying to cross from Russia to Georgia. I expect that the opposite direction will be simpler, although it can also be that all the border guards are occupied with people coming from Russia. So you may have to wait even if there is nobody else coming from Georgia.
A bigger problem may be accommodation. I think that accommodation on both sides of the border is very expensive now, and may not even be available at all.
At the same time, I think you can find some advantages in the current situation. For example, you may easily find transportation to the border in Georgia, and from the border in Russia, because the demand is very high in the opposite direction. Taxis and buses apparently go almost empty in your direction. So overall I think one may say that currently the situation is even more favorable for crossing into Russia than it usually was.
However, you may face problems coming back. I really doubt that Russia will close the border to foreigners, but the lines may be horribly long (think several days!)
At the same time, your statement "most inbound international flights to Russia have stopped" is not quite true. The flights from Europe and USA have stopped. But there are plenty of flights from Turkey, UAE, Armenia, Kazakhstan, and many other other countries. Moreover, I think these flights inbound to Russia should be rather cheap now because there is high demand on flights from Russia (and prices are very high). So the airlines may be selling flights to Russia very cheap because they need to move the aircraft there anyway.
If you give more details about where you need to travel to and what passport you have (as a separate question), I think people here will be glad to find alternate routes.