According to Wikipedia Rome is set from 52 BCE to 30 BCE. So I'll focus on that time period.

When Romans talked of "Asia" they almost exclusively meant Asia Minor, what makes up most of modern day Turkey. In particular they meant the Province of Asia, which took up the western third or so of Anatolia, as shown in this map.

That area was largely made up of what had been the Kingdom of Pergamum. Pergamum had been a wealthy independent Greek state. And the Pergamines had been Roman allies in the East for a long time. So in 133 when Attalus II died without a clear heir, he willed his kingdom to the Romans. The Kingdom of Pergamum became the Roman province of Asia. It was also reputably the wealthiest province in the Republic. Prior to that the area had been the ancient kingdom of Lydia, the kingdom of Croesus. The wealth of Lydia was so great that, "Rich as Croesus" is a proverb for unimaginable wealth. This map shows Lydia right before the rise of the Achaemenid Persian Empire.

Gaining Pergamum drew the Romans into a series of wars in Anatolia, first against the Seleucids and later against Pontus. Both the Seleucids and the Pontiacs were also famously wealthy. The most recent of these wars took place in 63 BCE, 10 years before the TV show starts, when Pompey the Great fought and defeated Mithridates IV in the Third Mithridatic War. That war brought fantastic wealth to Rome, and probably made Pompey the wealthiest man in Rome, rivaled only perhaps by Crassus.

The development of Roman influence in the area can be seen in this series of maps.

So so for a Roman to refer to a wealthy King of Asia around 40 BCE, he may have been referring to the legendary wealth of Croesus of Lydia, or the Kings of Pergamum, Pontus, or Seleucia. Or he may have been referring obliquely to Pompey who held the loyalty of millions of clients in the area. Or most likely all of the above, Asia was the place of famously wealthy kings after all.