Of the civs involved, Egypt takes it hands down.
The earliest other candidates are China, Assyria, Babylon, India, and Greece, and none of them come close in any way.
In terms of archaeologically attested cultures (@reddishrecue), Egypt's Naqada culture [which isnt the earliest, but its perhaps the most substantial pre-dynastic culture] dates from 4200ish BCE, China does seem to extend earlier and earlier with successive excavations (that do try and verify the Xia dynasty), but at best it is no earlier than 2700 BCE so far, Babylon was pretty much founded out of nothing by Hammurrabi's dynasty in the early second millennia BCE, Assyria (when it traded with southern anatolia) really doesnt extend earlier than the 2nd millennium BCE, India has the IVC (Indus Valley Culture) around 3300 BCE (even though that is clearly not the same as modern India), and Greece has Mycenae and Knossos before the 2nd millennium BCE.
Even if you ONLY go by the founding of a unified state, Egypt comes in around 3100 BCE, China *officially* comes in at 221 BCE, though i will concede that some sort of state existed as early as the Shang period (1600-1046 BCE) , Babylon comes about around 1800 BCE, Assyria comes in in the early 2nd millennium as the Mitanni collapsed, India comes in during the 4th/3rd centuries BCE (Mauryans), and Greece comes in (arguably) at 338 BCE (Chaeronea).
If you want to go by ruler, again, Egypt takes it. Ramesses II (1279-1213), while late by Egyptian standards, is still 2000 years before Wu, 700 years before Nebuchadrezzar II, 600 years before Assurbanipal, 3100 years before Gandhi, and 900 years before Alexander III.
As others have said, Sumeria would have predated egypt (and every other potential candidate) in these regards, but its not in Civ5.
And as for reddishrecue, if we look at all mammalian existence (including the Olmecs) as a single "culture", then yes, he is correct that the Olmecs were on Pangaea. However, if not, we can only hope that a Tardis is found and throws any sense of chronology out the window.