If so, how would they irrigate the crops? And if not, how did people live in such areas back then?
Sure! But it depends on lotsa stuff.
Irrigation systems depend on the technological level of sophistication, the ease of access to non-salinated water supplies (is there a river nearby?), the political and sociological organization of the culture, and agricultural knowledge.
There are many examples of pre-Roman societies and their use of irrigation. See: http://mygeologypage.ucdavis.edu/cowen/~gel115/115ch17oldirrigation.html -and- https://www.cabrillo.edu/~crsmith/southwest.html (The second link gives information the ways irrigation was accomplished.)
And if not...
In cases where irrigation would be impossible, economically unfeasible, or technologically too difficult, cultures were mostly nomadic and/or hunter-gatherers. Pastoral cultures (herding) still depend on water, but occasional or oasis-type watering holes might work to provide for the beasts.
EDIT: Also, the Romans weren't—by any stretch of the imagination—the first to invent irrigation systems. However, I'm guessing you mean "pre Iron Age."
Irrigation is at least 5,000 years old, predating the Roman empire by thousands of years.
For a concrete example, look at the Egyptians. Egypt is mostly a desert, with a big river running through it. According to Wikipedia, there's evidence they used irrigation about 3,800 years ago. That's hundreds of years before Rome was founded. (Rome itself was founded more than 700 years before the beginning of the Roman Empire).
You don't have to have a culture that's technically advanced to practice irrigation. For instance, the ancestral Puebloan peoples of the southwestern US practiced irrigation. In particular, the Hohokam built an extensive system around the area that's now Phoenix: http://www.waterhistory.org/histories/hohokam2/ I don't see any reason why Eurasian people couldn't have done similar things long before the Romans, or FTM the Egyptians.