She would have learned the Modern Arabic for written/formal purposes, but the spoken dialect she learned would have been the Levantine group (that would work across Syria, Lebanon, Palestine, etc). That version of Arabic is different than its close cousin, Mesopotamian (as in Iraqi). My understanding is that while Syrians might have difficulty understanding Iraqis, it's more akin to English spoken in U.S. vs. U.K. vs. Australia.
Now the North African dialects (Morocco, Algeria, Tunisia) are a whole other matter and would be a lot harder to understand... apparently no one in the Middle East really understands them. Egyptian Arabic is also tough for the others, but not as much as the other N African ones. The Gulf states are all pretty close and relatively easy to understand.
So my thought is that Elizabeth would have learned the Levantine version in school (while studying abroad in Morocco, she only focused on French, but liked the sound of Arabic) and picked up the Iraqi version while she was there for 4 years. She is, after all, described as being good at languages. In one of the early chapters, I mentioned that she had passable knowledge of Farsi as well, and I imagine she would try to pick up Hebrew while in Israel. I have a couple of friends like that, who can start conversing in difficult languages (like Russian) while studying there for a few months. So didn't seem that out of bounds when I was conceiving this character.
Hope that helps explain my thinking! I don't speak any Arabic, so this was all a best guess and if any native speakers find it wrong, then I will just beg artistic license.
