Focusing on Europe and North America:

Distilled liquor starts to gain more than a microscopic presence in Europe at the very end of the Middle Ages. However, at this point, it's mostly in the hands of apothecaries prescribing it as medication.

It's also very limited in consumption in ways very important to our purposes. The strength of hard liquor did not go unnoticed by city and imperial authorities, who (a) instituted wildly high prices to discourage significant purchase (b) actually limited amounts that could be purchased. In particular: in some places/at some times, customers of apothecaries and grocers were limited to purchasing brandy in-store, for immediate consumption standing at the counter.

But here's the really interesting thing. "Shot" as an alcohol-related word, in the 1500s, still refers to the bar tab (and in fact, for the group, not even for an individual). But "dram," another modern word for...well, a shot--dram seems to refer to a doseage of alcohol-medicine by the end of the 16th century. The OED has a written reference from 1590, suggesting that a dram had been better than a damn years before.

So we're still at liquor as medicine, but the idea of drinking a shot and calling it by a word that will later come to mean our modern shot, is already present.

But as people are gaining experience with quickly drinking small amounts of distilled liquor, a second practice is becoming quite popular. One that involves swift consumption of an alcoholic drink in ways that would hopefully limit how much you would drink at once.

Around 1500+, the practice of "pledging healths" becomes not just wildly popular but seen as a genuine social problem. Which is to say: drinking toasts, but turning it into a drinking game and a competitive sport. People would go around the table pledging a health to which everyone would have to drink. At the beginning of the night, the point was to pledge the most clever health (a backhanded compliment to your friend's wife, perhaps?); at the end of the night, "most clever" probably had a...rather more relaxed meaning.

Now, pledging healths in 1517 is not drinking shots in our modern sense. But it's a step towards that sort of practice.

Different scholars will point to different points in the next 200 or so years when drinking distilled liquor becomes truly popular. Ann Tlusty lays out laws from Augsburg and elsewhere in the Holy Roman Empire at the end of the sixteenth and early sixteenth centuries that get increasingly shrill about trying to limit the drinking of hard liquor, especially brandy. (Although much of this also involves the sale of watered-down or low-quality liquor, which would either be less likely to be drunk as shots for the obviously reason, or more likely if it tasted gross).

Jack Blocker, on the other hand, puts the time frame just a slight bit later, and suggests it is especially with European colonialism in the West Indies and North America where white people become especially fond of drinking rum. Yes, very weak alcohol consumption was extremely popular in homes at this time--but it was beer. Rum and other drinks were more limited to public houses, and apparently these were pretty seedy on the whole. (In other words, you're not going to have men in robes with pipes discussing international affairs while sipping on gin and tonic, with their pinkies in the air.)

By the early 18th century, "dram" (at the very least) seems to signify a shot in the modern sense. In one of his satires, Alexander Pope has a character mistake someone saying drama for dram, and explain apologetically that there is no jenever left. I don't drink, but I am informed that jenever is a form of gin typically consumed in quick, small amounts.

So by your 1880s ball, my gentlefriend, you would definitely not be throwing away your shot.