Complete Spread of Buddhism into the European powers (Greece or Rome) would give you a Buddhist Syria-Palestina, Mesopotamia and Egypt+North Africa. I doubt a separate spread of Buddhism into Arabia from India and an Islam like expansion is possible, since that happened due to the fact that both empires were exhausted and turned their borders into a near wasteland, in the 7th Century, and not to mention the Migration Age that had battered Rome.
Both of these are unlikely to happen at an earlier era when Buddhism could spread as Rome and Persia were at their Zenith, back then. The best option for spread is hence, through Europe, which was more conducive for a Conversion to Buddhism as they had many Pagan religions and an integration of Buddhism is easy. Zoroastrian Persia already had a structured Monotheistic religion with strong doctrines and philosophies, already, and may be impossible to convert, unless under a converted Seleucid or a Bactrian Greek annexation. So Persia could remain a patch while the lands around it, from West Africa to Kazakhstan could be Buddhist.