you'd need a facility with a total volume of 18,600,000 m^3
No, you'd need to process that much air. The facility could be much smaller, but you'd need to move so much air through it you'd be expending much more energy moving the air than you'd get back from the methanol.
Even if you contain all the air in the one facility, you won't be using 18 million cubic meters of water and catalyst, so you still have to move the air.
But it is a beautiful thought experiment to think about such a CO2 to methanol recovery device on the output of the burner of a hot air balloon. Maybe you could make enough methanol from it that you could replace the tank of propane and burn only methanol. And with thermal electric generation, you could install wireless networking in the balloon and have permanent access points in the sky...