Hi there, I've written about time zones a few times in the past. The short answer is that people who studied these things, at least, understood that local time would be different in different parts of the earth. To quote myself from this old answer:
Hipparchus of Nicaea (c. ~190-120 BCE) seems to have been the first person to propose using a grid system to find the position of cities (and other places) on a globe, which implies an understanding of longitude. He built on earlier work by Eratosthenes of Cyrene (c. 276-194 BCE), who had mapped the known Earth, including finding its circumference. Hipparchus' method of finding the longitude of places was to use the differences in timing of lunar eclipses at different points on the globe to calculate the difference between local time of those points; the drawback is that there was no accurate-enough method of timekeeping to lead to useful calculations. Edit to clarify: The difference in local time between observed beginning and end of the eclipses would serve, essentially, as a way to understand the longitude between places.
Now that said, other than the theoretical implications of time being different in different places (determining the earth's size, for example) travel speeds meant that the idea of time zones was interesting, but had little practical utility before the invention of the railroad -- even travel at sea, for example, rarely moved quickly enough for there to be any discernible "ship lag" until the very late 19th century. (For more about naval timekeeping, check this old thread out.) So your theoretical Silk Road traveler might have known that he earth was round -- at least by the Middle Ages, we have evidence of sermons offhandedly referring to a round Earth -- but wouldn't need to think about local time being different in other places on a practical level.
Here are some other old threads on time zones:
and an overall link drop to clean up anything I missed: https://www.reddit.com/r/AskHistorians/comments/4y6lez/when_did_people_first_realise_that_the_time_of/