This is a hard question to answer, because the word classical is debatable. In light of this I will only comment on classical Western cultures. This means that I will look at the Greeks and Romans. In terms of challenges, there were many. I will list four.
First, there was always the challenge of warfare. The ancient world did not have a version of the United Nations. So, there was incessant warfare. This fact brought much destruction. Hence, there was the challenge of survival. For example, think of the Peloponnesian War War, or the civil wars in Rome.
Second, there was always the challenge of government. The best form of government was always debated. For instance, Rome started off as a monarchy and it was overthrown and it became an Republic. Later on it turned into an Empire.
Third, there was the challenge of food. This is to say that ancient governments always had to procure food for its people. If they did not, there could be riots.
Finally, there was the problem of debt. We see this in the works of Solon, the ancient Greek lawgiver. We also see this in the works of Livy. There is a strong thesis that debt created may revolutions.