This is an easy one: the wikipedia entry for Cleopatra is flat out wrong. You are correct: even "Old Arabic" is not attested before the first century CE, and even if you want to push the very earliest hints back into the 2nd century BCE, it would be tiny groups in Arabia proper in the first, and Cleopatra would definitely have no cause to speak it or even know of its existence.

The wikipedia sentence cites a sourcebook by Jones for this, which in turn cites Plutarch, who wrote a "biography" of the queen. He says:

Pleasure also came with the tone of her voice; and her tongue was like a many-stringed instrument: she could turn it easily to whichever language she wished and she conversed with few barbarians entirely through an interpreter, and she gave her decisions herself to most of them, including Ethiopians, Troglodytes, Hebrews, Arabians, Syrians, Medes, and Parthians.

"Arabic" is not the same as talking to "Arabians" in the 1st century BCE. Whoever wrote that page should probably edit that bit. Plutarch is also not exactly a reliable source for little details like this, though other sources also mention Cleopatra's linguistic skills. She did not apparently speak Latin, however, and seduced Caesar and later Marcus Antonius in her native Greek!