I'm not sure train is practical either as I don't think there are any functioning railways in Lebanon. You would need to travel from Sochi through Georgia and Turkey and there might (or might not) be bus services across some of those borders. Russia and Georgia are somewhere close to at war so the border may be closed.
You may need visas for these countries.
Is it possible to get a train to some bigger city not too far away that has a direct flight to Turkey?
You can fly from Sochi to Istanbul on Donavia airline (a subsidiary of Aeroflot), 1.45 minutes flight, they don't fly everyday, but at least 3 times a week, and then with perhaps Turkish airlines to Beirut.
This would be the best option, but since on 2 separate tickets and probably one way, I would expect to pay ov er 600 USD for the routes.
Train, even if possible, would take forever and would require you to get a number of visas.
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