The time frame would be around 60 to 30 B.C.
The Arabic world was we conceive of it didn't exist yet and wouldn't until the Muslim conquests of the 7th and 8th centuries.
These reddit threads may help, although others should chime in if they feel like it.
As pointed out, the Arabic world [ETA: in this timeframe] is not terribly united and does not bear much resemblance to its post-Islam form, but certainly there were things going on in the Arabian peninsula that had some bearing on the Roman world. The Romans in the time frame of the show were in direct control of portions of the Eastern Mediterranean, and had a tremendous degree of influence over the Ptolemaic kingdom in Egypt. There was oriental trade taking place through the Arabian peninsula, and kingdoms such as the Nabataeans (famous for the ancient city of Petra which you have probably seen pictures of, and which was used as a set for an Indiana Jones film) in roughly modern Jordan grew wealthy off of this trade. Much of what is now dominated by Arab culture and language was not at the time -- Egypt was still culturally distinct and ruled by a Macedonian family that had, for example, mixed native Egyptian religion with their own, to horrendously over-simplify the situation. Mesopotamia was dominated by the Parthian Empire, which had been in contact with Rome since at least Sulla and would later become a rather significant rival of Rome, a sort of resurgence of the old Persian empire. There were also very wealthy trade kingdoms in the milder climate of the southern Arabian peninsula, modern Yemen, such as Qataban, which had access to Indian Ocean trade and were a source for some exotic goods that fetched a high price in Rome, especially incenses like frankincense and myrrh.
The Ptolemies -- the descendants of Ptolemy I, Alexander's boyhood friend -- ruled in Egypt at the time. Cleopatra was a Ptolemaic ruler, as was the relation she was fighting with over the throne. They're a fascinating group. Ptolemy I took the name "Soter," or "Savior," when he took over Egypt after Alexander's death; he ruled in the style of Macedonian kings but with a strong overlay of Egyptian imagery, which he immediately adopted. He styled himself and his descendants as heirs of the Pharaonic religious tradition. We refer to his society -- and to those created by the descendants of other Alexandrian diadochi ("successors") -- as a Hellenistic monarchy. They span the time between appx 300 BCE to the rise of the Roman Empire.
Hellenistic successor states waged almost constant warfare amongst themselves, especially in the early years. In fact, the Colossus of Rhodes was built to commemorate Rhodes's victory over one successor, with the help of another (the Ptolemies). Each state relied heavily on mercenary armies, Ptolemy more than others. They adopted Greek notions of city planning, among others.
In these and other ways, the societies, Egypt included, became heavily influenced ("Hellenized") by Greek ideas and culture. It's this that Caesar would have encountered in Cleopatra -- a Greek-styled family ruling a Hellenized Egypt under color of Egyptian norms. (Source: Companion to the Hellenistic World, among others.)
