Since the Assyrian Empire is not well-known to virtually anyone except ancient historians, qualifying the popularity of a ruler may be hard. There were no Assyrian rulers whose names are still well-known today, a contrast to other ancient leaders like Julius Caesar or King Tut.

Perhaps the most famous ruler was Sargon II. He is remembered as a great king of Assyria, living in the 8th century BCE; like many other kings his name is known today from the Hebrew Bible. In fact, the name "Sargon" is a Hebrew transliteration; Sargon's true name was Sharru-kin. Sargon is famous for his campaigns in the Zagros Mountains in northern Iraq, pacifying rebels and bringing in new territory to his empire. We have a record of this campaign, one of the earliest surviving records of warfare.

Additionally, Sargon helped to popularize writing on wax tablets. This form of writing was still very much in its infancy: Sargon's decision to record more and more political decisions to wax has helped historians to better understand and contextualize his life. Finally, Sargon managed to build a mighty palace at a place called Dur Sharrukin, although he died shortly after.

If we are talking about fame in the non-academic world, I would argue that the most famous ruler of the Assyrians was Sennacherib II.  Sennacherib is famous for two reasons.  First, his wars against Judah are mentioned in the Bible.   Second, he is mentioned in a famous poem by Lord Byron called "The Destruction of Sennacherib."  These sources make him better-known to the average person than most rulers of Assyria.

In history books, the most famous Assyrian rulers are Tiglath-pileser I and Ashurbanipal.  Tiglath-pileser is credited with being the first to truly expand Assyrian Power and to create a true empire.  Ashurbanipal is famous for being one of the strongest of the Assyrian rulers, but also one of the last to reign over the empire before it began to fall apart.