This is the most elusive marshrutka in Georgia. I asked a taxi-driver and he said there were no direct bus to Mestia. And I talked with a minibus driver who worked at Batumi bus station and other locals and get the same answer - 'No'. But! While being in Mestia I saw it! It was a typical for Georgia minibus with a sign 'Batumi' and a phone number (look at the photo above).
So if you can't speak Georgian or Russian, ask the owner of your accommodation to call to ask the schedule.
How much does it cost to backpack in Georgia? - if you still have any questions like that, read a guide written by a traveler who spent months in Georgia - 'Backpacking on a low budget in Georgia (country) - how much is it?'