Probably. There's no such thing as The Common Cold: today there's four most common viruses that cause cold-like disease, each of which has multiple subtypes, and a bunch of less-common causes of cold-like symptoms(1). We only identified rhinovirus, which is the most common cause of colds, in the 1950s. Coronaviruses (yes, the same type of virus currently involved in the epidemic in China) were identified later than that.

Unlike the flu, which tends to change every year, there's little evidence of rhinoviruses replacing themselves with "better" versions(2). There are many variations, but they tend to remain as such. I don't know of any evidence that any particular serotype of rhinovirus was around [i]x[/i] number of years ago.

With that said, there's significant evidence that people routinely experienced the same set of symptoms we would consider a cold as far back as we have recordings about medicine. The Ebers Papyrus, which was written in Egypt probably before the 16th century BCE, describes ways of dealing with all sorts of medical ailments, including a cough and a runny nose.

As for the Greeks, they absolutely had colds. Galen, who lived about 200 years BCE, identified a runny/stuffy nose as basically a waste product and not an indicator of serious illness (or, as he would have framed it, a serious imbalance of the four humors). He, and his predecessor Hippocrates (of the Hippocratic Oath), both identified seasonal variations in illnesses, and identified cold-like symptoms as more common in the winter.

Much of the above is summarized from "A short history of the common cold" by Isabel Atzl and Roland Helms, the first chapter of a book called "The Common Cold". Almost all of the first chapter is available here.