In which month was the Pur (the lot) cast before Haman?

Answer: Nisan

In Nisan, the first month of Xerxes' twelfth year, the lot was cast; it fell on Adar, the twelfth month.

“In the twelfth year of King Xerxes, in the first month, the month of Nisan, the Pur (that is, the lot) was cast before Haman to determine a day and month. And the lot fell on the twelfth month, the month of Adar.”— Esther 3:7 (BSB)