Well having a Greek Arabia is simple. Have Alexander live a little longer. He was about to go on campaign in Arabia when he fell ill and soon died.
As for a Roman Arabia, I remember hearing somewhere about an attempted Roman campaign in the region.
Okay, after looking it up on google, I found it. It was the Aelius Gallus expedition from 26-24 BC. Gallus was the praefect of Egypt.
According to Wikipedia, " His expedition to Arabia Felix, of which an account is given by his friend[citation needed] Strabo,[3] as well as by Cassius Dio[4] and Pliny the Elder[5][6] turned out to be a complete failure."
http://en.wikipedia.org/wiki/Romans_in_Arabia
So, to have a Roman arabia, have it succeed. Or have Augustus or someone else feel it necessesary to go into Arabia.
Also, the Romans had a province called
Arabia Petraea.