Time zones are a 19th-century invention meant to standardize the differences in local solar time (i.e., where noon is defined as the moment of the Sun's zenith) that you will see on different parts of the Earth. You only need "time zones" when you are traveling fast enough that these differences have a serious practical effect (they were invented by railway companies who found that if everyone uses their own definition of "noon" then you can have real problems in sharing tracks). The fact that time zones are invented can be easily seen by looking at them — they don't actually track with solar time all that well in many places, for political reasons (China being an obvious and glaring example of this).

The Ancients definitely understood that solar time varied depending on your longitude. This is a rather trivial observation if you think about it (it is a consequence of living on a sphere), and one that you can confirm with any semi-rigorous astronomical observation (e.g., by comparing the time at which astronomers in different parts of the country observe something like a lunar eclipse). Medieval European astronomers would have understood this as well, as they based their astronomical understanding primarily on those same Ancient texts (like Ptolemy's Almagest).

That does not mean that one could not still be surprised by the quirkiest aspect of this phenomena: Magellan's crew expressed great shock that they "lost a day" while circumnavigating the world (they kept very close track of the days in their ship's log), even though that is a consequence of the same effect (but one that continues to boggle the minds of even modern people, as anyone who has traveled across the international date line can attest).