Before the Islamic conquest of Syria and the whole Levant, how widespread was the use of the Arabic language there? Are there any resources show what the percentage of Arabic in that region is as well?
To address Mark C Wallace's point, being ethnically Arabic and speaking the Arabic language are not the same thing. Arabic people existed long before the language we now call Arabic developed, albeit we tend not to refer to them as Arabic until the era of the spread of Islam and the Arabic language.
This is not a detailed answer, and it relies on Wikipedia (https://en.wikipedia.org/wiki/Syria#Middle_Ages). But for what it's worth:
The predominant languages spoken in Syria before the conquest by the Arab Rashidun army in 640 was Greek and Aramaic, reflecting the influence of the Byzantine empire and Byzantine Christianity. With the conquest, Islam became the dominant religion at the same time as Arabic became the dominant language.
The second part of your question I am unclear about. "Are there any resources show what is the percent of Arab in that region as well?"
Do you mean the percentage of the population that spoke Arabic before the conquest? Or today? Or are you talking about people of ethnically Arabic origin? If you are talking about medieval Syria, my guess would be 'no'. Detailed statistics for medieval history seldom/never exist. My best guess would be that Syria before the conquest was already ethnically 'Arabic' (except in this period what today would be called 'Arabic' was called Assyrian, Phoenician or Persian), but linguistically and culturally Byzantine. The Ghassanid Arabic group emigrated to Syria and the Levant in the 3rd century. In other words, the ethnicity that today we call Arabic, is simply the modern term for the same ethnicity that was called by a variety of names before the advent of Islam. After the conquest it became linguistically and culturally 'Arabic' too.
But I am far out of my area of expertise.
So in short, the answer to your question 'how popular was Arabic in the Levant before the conquest of Islam' is 'not very'.
Greek was the language of the ruling elite and Aramaic (or "Syriac") was widespread among the peasantry. However, Arabic was also widely spoken throughout Greater Syria, since there were many Arabic tribes in the area as well as in Iraq. Arabic inscriptions dating many centuries before Islam are found throughout Syria-Palestine, especially the southern parts. There were several Arab kingdoms as well, the Ghassanids (centered south of Damascus) and Lakhmids (of southern Iraq) being the last of these before the Islamic conquests.
Hoyland, Robert, Arabia and the Arabs: From the Bronze Age to the Coming of Islam
Shahid, Irfan, Rome and the Arabs and his multi-volume Byzantium and the Arabs
Macdonald, Michael C.A., Literacy and Identity in Pre-Islamic Arabia (Variorum Collected Studies Series)
A collection of inscriptions with photographs or facsimiles.
The answers above attempt to simply ignore history!
Arab tribes emigrated from Southern Arabia due to various adverse climatic conditions and growth in population which could not be supported with the grazing land in Southern Arabia and the coastal strip running on the eastern coast of the Red Sea.
The first target of immigration was the Gulf in the East. Many Arab tribes sought that location like the Tenoukhis,for example.
But even there the need to move presented itself and the move was northerly, into Mesopotamia proper and North-Westerly as far as the boarders separating modern Syria from Turkey.
Suffice it to say that battles waged by an Arab tribe back in 850 BC, suggests totally different from what Western historians present to us - without the slightest archaeological or anthropological proof.
Remember: the Arabs are "inferior" people as far as the cultural structure of most Western societies. Back since the 18th Century, a dirty beggar wearing torn clothes was known in England as the "Street Arab."
I am a mathematician and not an historian, but have landed on much achievements of the Arabs in Andalusia in the 8th-15th Century. Whatever they did in Medicine, Chemistry, Physics, Astronomy, and even Aeronautics, was ignored - apart from the Arabic translation of the Greek works... These were translated from Greek and used by the Arabs in Syria and Andalusia as basis for their advances even in philosophy. The books were translated into Latin by Jewish refugees from the murderous Papal inquisition in Andalusia.
That is, Mediveal Europe went back top the same starting point the Arabs had 600 years before...
Small wonder, it took the American Optical Association 700 years to get to grips with Ibn Al-Haytham extensive works on optics, to award him the title, "The Father of Optics," but keep this down, we do not want a revision of the lowly stature of Arabs in the US and the West in general.
A former graduate student of mine, now lecturing in a university in Washington State, wrote asking me for information about "Ibn Firnas." I responded with all what I know, but asked for the reason of the question... Well, it was the renaming of the biggest aero lab in her university, "The Ibn Firnas" Laboratory.
Remember Ibn Firnas. The one who designed the first human aerofoil (Wings), and documented the design and dynamics of the birds wings in 1120 AD, and did glide from the Minaret safely to the ground wearing his invention...
But keep this down too.
Well, there is much, too much to discover about the Arabs. All what I can say is that I start with the assumption that the Arabs were in the Levant much, much earlier than is supposed... It would be nice if they work on the DNA of the Cannanites, the Phoenicians and other groups of people who appear in the Western history to have cropped up form nowhere!