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The Arab tribes were spread out widely, and the land of Arabia was not as viable to live on as the Fertile Crescent, so I would expect the population to be lower. But, how much in an estimated range was the population exactly before Islam and its conquests according to historians?
As a related question, how many people spoke Arabic at the time in and outside of Arabia?
As a follow up question, how much of a role did the population play in the ability of the Arabs to conquer what they did?
According to Colin McEvedy and Richard Jones, at the beginning of the Christian era it was about 2 million, and in the next 2 pairs of centuries it geometrically rose to about 2 2/3 million, then nearly 4. It had climbed to a high of about five and a quarter million around the time of Muhammad, a high-water mark that was receded from a bit to about 4.5 million within the first two Islamic centuries, not to be reached again until the Modern Era.
For this entire period, about half of this amount they place in The Yemen, the balance of the other half in the interior, and only trivial numbers on the Gulf coast and Oman. However, large amounts of the half in Yemen would not have spoken Arabic, but rather an assortment of related South Arabian (Semitic) languages.
For the region, these weren't gigantic numbers, but they were very respectable, and it was probably rather a lot for them. On population alone the Arabian peninsula was in the same league as the contemporaneous existing power centers in the boundaries of what today are Turkey and Persia*. McEvedy and Jones' text speculates that the high numbers likely put strain on the resources of this relatively poorly-resourced area, and that might partially explain its seemingly sudden foreign adventurism.
In Colin's excellent New Pengiun Atlas of Medieval History, he points out that the relatively easy victories the early Islamic armies had in the region may have been a reflection of the inhabitants not having a lot of fondness for the Greeks and Persians who had been fighting over their territory for centuries. Exploring this idea further, its worth noting that even if the locals didn't necessarily speak Arabic, Arabic is a Afroasiatic language, and such languages at the time were the language of the common people across the Levant and north Africa. So there was certainly at least some level of cultural affinity with Arabs that there would not have been with Greeks and Persians (both speaking Indo-European languages).
Pre-Islamic distribution of Afroasiatic language families
Image taken from Expansion of Afroasiatic Languages video on Youtube, based "loosely" on the work of Harold Flemming.
Distribution of Semitic Languages
* - The boundaries of modern Turkey at that time they put at around 6 million, and Iran at about 5.
You have a problem in the history of Yemen and southern Arabia in general. The information you have is very old: first, the archaeological Soviet mission discovered evidence of human activity in Hadhramaut dating back a million and a half years. but now dating back 2.6 millions as British Encyclopedia say. Secondly, the southern Arab Musnad line goes back four thousand years, and perhaps even further, according to the US-Canadian joint mission in Yemen. Third, Yemen is the closest geographical point to East Africa, which is supposed to be the origin of mankind.
The traces of the Yemeni civilization in East Africa are present and strong. In contrast, the amount of Yemeni antiquities and inscriptions reaches tens of thousands (refer to the DASI International Code). As a simple example, the (al Maqar ) civilization in southern Saudi Arabia (all of the Saudi south was subordinate to Yemen) dates back ten thousand years. And there are no traces of foreign civilizations at all in Yemen.
All the tribes in East Africa and in the Arab world recognize their Hamiri South Arab origin. Arab and African historians are unanimous on that. Dr. Arnold Toynbee, in his book History of Humanity, attests that southern Arabia is the vital home of Apicomina in the ancient world. Back in the ancient South Arabian inscriptions, there are statistics of the number of dead local war battles and prisoners in the thousands, indicating the density of the population there. Yemen has been known since ancient times for migrations to the north and east.
When the Prophet Muhammad appeared, the Yemeni tribes were at the forefront of the Arab armies and most of them until they reached Andalusia in Spain. Yemen is the origin of the Arabs and the southern Arabic language. Because present-day Arabic is Northern Arabia, and it is modern since only six 1500 years. Conclusion Yemen is in too many secrets. Only 10% of it was revealed due to the Saudi-Emirati occupation of Yemen. Therefore, I expect that the population of southern Arabia was not less than 5 million people, at least ten thousand years ago, if not more.
