When did Judaism become popular in the ancient world, and what caused it to spread faster at that point in time? (ex: Christianity had its spread assisted by Emperor Constantine in the Roman Empire)
Around the beginning of the Christian era there were people called “God fearers”, who were gentiles (non-Jews) who worshiped the God of the Jews, but were not circumcised and did not follow the Law of Moses. (See for example Acts 13:16 and 13:26). These “God fearers” seem to have been the prime target of Paul’s preaching. They should be seen in the context of the fad for “Oriental religions” (Mithraism, Isis-worship and the like) in the Roman Empire.
The first relatively wide knowledge of Jews and the Jew religion came as a result of the Babylon enslaving. 536 BC.
Judaism became widely known in the Christian times, after wide spreading of Jews as the result of the last Jew-Roman war. 135 AC.
But the times when several non-Jew nations accepted the Judaism came even later - about 7-8 cent AC. Khasars are the most known example.
Judaism never become popular like the large world religions (Christianity, Budism, Islam...etc) as it does not have a universal message. Judaism was, until very recently, a religion by birth that did not proselytize. It is a system of laws intended for a chosen people, not all people.
Chritianity, in its earliest days, was just a sect of Judaism, without a specifically universal message. It was Paul that laid the groundwork to make Chritianity a call to all people.