It seems like unless you could instantaneously communicate (via phone or something) with someone in a different time zone, it would be very difficult to tell.
The ancient Egyptians were aware that the earth was a sphere. They had created a 365 day solar calendar pretty early on in their history as they needed it in order to predict the flooding of the Nile river. I would assume that this knowledge would lead them to believe that different parts of the world saw the sun at different times.
There was a Greek mathematician named Eratosthenes who calculated the circumference of the earth around 240 B.C.E. He did this using the distance between 2 cities and measured the angles of the sun's rays during a solstice. Its a pretty neat experiment considering the time period it was done in. I mention it because it was probably pretty clear at this point that different places experienced the day/night cycle at different times.
Claudius Ptolemy, who lived between 90 - 168 C.E., created a geocentric model of the solar system. (The model where everything revolves around the earth.) While it was wrong, it does correctly explain the day/night cycle. It wasn't the first model of this type, but his version is the most famous and was widely accepted until the 1500's.
I don't have any specific sources on hand, but finding information on these particular people isn't too difficult.
Others have posted links to the discussion of time zones, but the notion of traveling fast enough for time zones to make a difference was still novel enough in 1873 that Jules Verne was actually able to use it as a plot device in Around the World in Eighty Days. Given that, it can be argued that the average well read person was well aware of the concept intellectually, but had likely not experienced "jet lag" to any extent that would spoil the surprise twist at the end of the story.
If you don't know what the twist is, I won't spoil it. You can read it here or just skip to the end of the plot summary to find out.
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Related question: what about with seasons? When did people realize the Southern Hemisphere had opposite seasons from the Northern Hemisphere?