When was The Wailing Wall built?

There is no such thing as a wailing wall. This is a misnomer imposed by non-Jewish people who misunderstood the distinctive style of Jewish prayer, and thought it was wailing.

You are most likely referring to the Western Wall in Jerusalem.

The Western Wall was built as part of the expansion of the 2nd Jewish Temple, approximate 20 BCE. It is a retaining wall of the Temple mount that was built by King Herod.