Also, did big alcohol manufacturers have the same amount of influence in those ancient cultures as companies like Anheuser Busch have today?

There really weren't "big alcohol manufacturers," as we would think of them today, in the ancient world, for a couple of reasons. First of all, the ancients, even the Romans who were highly advanced logistical geniuses compared to their worldwide contemporaries, did not really understand the ins and outs of the assembly-line process. In particular, they did not understand the idea of replaceable-exchangeable parts and mechanization the way we do today. So, even the big vineyards (the Romans didn't really have distilleries, although there is some evidence they tried to distill wine to more powerful forms) couldn't produce the kinds of volume, even relatively speaking, that would be necessary to centralize a single brand for a marketing effort of the kind you're speaking of.

Another issue is that Greeks and Romans did not have the formalized legal fiction so critical to our understanding of business today: limited liability. I have argued in several articles I've written (outside of Reddit), that no single intangible concept has more vastly shaped the modern world than this one, and without it, the Romans, for example, would simply not have been able to bear the risk involved in creating a massive, Empire-wide alcohol corporation (the very world "corporation" is tied to the concept itself), which, again, makes branding and marketing difficult.

Another issue is there weren't the same kinds of channels (literally). Romans didn't have mass media. But, they did have the famous panem et circenses. And this was often used as PR for political figures. These figures would give away free "schwag" (in the form of food and entertainment) to the hoi polloi, in exchange for votes or goodwill, and that was a form of advertising. Soldiers would get a ration of wine (sometimes mixed with water or vinegar) to keep them happy. And yes, wine itself was sometimes advertised by shops in towns on signs, via graffiti, by criers, or through word-of-mouth marketing. But this would've been more generalized: "Boy, wine sure is tasty, and Caecilius' shop has the best wine!" not "I'm Titus the Gladiator, and when I get done banging 37 hot Thracian slave girls, I always drink Julian Wine!"

You also have to remember that the average person today, even the average really poor person, is way, way, way, WAY richer, than even the average middle-class person a long time ago. It makes sense to advertise to us, as we have a lot of disposable income, relatively speaking. Many people a long time ago were slaves or subsistence farmers. Advertising to a broad base of people would've done very little good. They didn't have any extra money to spend on anything, let alone wine. This is the same reason there wasn't much advertising in Communist Russia, either, except for state propaganda -- nobody had any choice in what they could buy, and nobody had the spare vouchers or whatever to buy extra stuff even if they had a choice.

So, apart from lacking the ability and logistics to supply the product, the channels and understanding to advertise, the other problem was there was, essentially no demand to advertise to, either.

Sorry, this was a little all over the place, but I'm still recovering from the Super Bowl festivities myself. Sources and more depth available upon request, but the general answer is: no, advertising as such didn't exist in classical times.

In Pompeii, there is a painted inscription next to a wine shop saying "Vesuvino," a pun on the nearby Mount Vesuvius and the Latin word for wine. Not exactly what you're looking for, but certainly an example of marketing from the ancient world.