They'd assimilate a sizable chunk of the lands immediate the desert in modern southern Iraq, Jordan, and Israel, which was an ongoing process in Late Antiquity. They could easily emerge as a formidable opponent to Rome and Persia, but would easily fragment due to lack of unity and internal disputes. Egypt will likely be conquered, probably a sizable amount of the Levant too. They almost certainly won't get further than Cyrenaica though, although they can certainly raid Africa/Numidia.
IMO the Turks are the best model. Like the Turks you'd easily see scattered populations of Arabs around the MENA but likely only a majority in the Arabian peninsula. I think they'd leave a huge linguistic influence on Coptic and Aramaic though, as much as Greek had at that point.