This Bible contradiction is from the Skeptic's Annotated Bible.
No, it was never rebuilt after Joshua destroyed it.
Joshua 8:28 View context
And Joshua burnt Ai, and made it an heap for ever, [even] a desolation unto this day.
Yes, it existed at the time of the Babylonian captivity.
Nehemiah 7:32 View context
The men of Beth-el and Ai, an hundred twenty and three.