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.
In the 1st century CE, a large fraction of the eastern part of the Roman empire was Jewish. Estimates vary widely, not to say wildly. According to Wikipedia:
According to Theodor Mommsen, in the first century C.E. there were no fewer than 1,000,000 Jews in Egypt, in a total of 8,000,000 inhabitants; of these 200,000 lived in Alexandria, whose total population was 500,000. Adolf Harnack (Ausbreitung des Christentums, Leipzig, 1902) reckons that there were 1,000,000 Jews in Syria (which included Lebanon) and the areas east of the Euphrates at the time of Nero in 60's CE, and 700,000 in Judea, and he allows for an additional 1,500,000 in other places, thus estimating that there were in the first century 4,200,000 Jews in the world. Jacobs remarks that this estimate is probably excessive.Historical Jewish Population Comparisons
That's out of a total estimated population of 20.9 M in the Greek Eastern part of the Roman Empire(Demography of the Roman Empire).
Even if that's off by a factor of 2, that's something like 10% of the population.
These Jews were targeted by Christian missionaries, and their numbers fell drastically during the following centuries.
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.
I would say that with the emergence of Ancient Alexandria and its famous Library, various cultures began to translate diverse works, including, The Old Testament. Around the 200's BC/BCE, The Old Testament was translated into Greek and was made available to a wider, non-Jewish audience.
However, it was really with the rise of Christianity that many of the ideas of Judaism were widely introduced into the pagan world, particularly by Paul of Tarsus.
While Judaism, as a religion, was much smaller, comparatively speaking-(in terms of population numbers), some of the ideas of Judaism became increasingly "popular" with the emergence of the early Christian community. These ideas included, the belief in a Messianic figure, Baptism-(though baptism did exist in other cultures, albeit in different forms), the concept of sin, the emphasis on praying to a single God, the abandonment of polytheism, as well as adhering to-(or at least having some orientation towards or basic knowledge of) The Ten Commandments. In a way, Paul of Tarsus helped to bring certain ideas of Judaism to the pagans through the introduction of Christianity. And as Christianity grew in many lands throughout the globe and evolved over the centuries, so too, did the knowledge (or at least a familiarity) with Judaism.
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.