When was the beginning and end of Old Babylon

It began in 1895 BC and ended in 539 BC

The Fall of Babylon denotes the end of the Neo-Babylonian Empire after it was conquered by the Achaemenid Empire in 539 BCE.

Nabonidus, son of the Assyrian priestess Adda-Guppi,came to the throne in 556 BCE, after overthrowing the young king Labashi-Marduk. For long periods he entrusted rule to his son, prince and coregent Belshazzar, who was a capable soldier, but a poor politician. All of this left him somewhat unpopular with many of his subjects, particularly the priesthood and the military class.To the east, the Achaemenid Empire had been growing in strength. In 539 BCE, Cyrus the Great invaded Babylonia, turning it into a satrapy of the Achaemenid Empire. Cyrus then claimed to be the legitimate successor of the ancient Babylonian kings and became popular in Babylon itself, in contrast to Nabonidus.

I hope this will help