I thought I'd research on this, but unfortunately no data is available.
I looked at Saudi's Central Department of Statistics & Information to compare the demographics of the country on a year by year basis, to see (a) how many Syrians are in KSA (b) how much of it is due to war and how much of it is normal acceptance of a labor force
استعراض - population statistics database - statistics and information database
Unfortunately, no statistics in the site has anything data that can support that
I found this, which mentions 1,000,000 Syrians, but again, how much is that due to war, since it also mentions 1 million Yemenis, 1 million Egyptians, 0.5-0.9 Sudanese, 2-2.8 m Indians, 1.5 million Pakistanis, 1.5 million Bangladeshi, 1.5 million Indonesian, and so on. However, they mention it all as estimates which means there is no clear source of demographic statistics.
While it's nice for a country to not accept people just as refugees (that is, put them in camps and treat them like shit), but keep in mind that, one needs to understand if Saudi took any initiative to support the Syrias in them going to KSA, or its just a normal procedure of them accepting a labor work force. That is, if KSA has been giving work visas to Bangladeshi, Pakistani, Indian, Egyptian, Syrian, etc, all these years to go to KSA & work, what exactly has changed for the Syrians?