The Bible doesn’t tell us much about the world before the Fall. But we can make a few inferences based on what God has given us.
So the Lord God caused a deep sleep  to fall upon the man, and he slept; then He took one of his ribs and closed up the flesh at that place. (Genesis 2:21)
The word used here means specifically a deep sleep (Hebrew: tardemah, Strong’s number 08639). This implies that normal sleep probably did exist, and this deep sleep is much more than normal sleep. So, I suspect Adam and Eve did have some sort of regular sleep cycle.
However, I wouldn’t be certain, as Adam and Eve could have been created merely with the capability to sleep initially. They may not have needed it. Perhaps sleep, after the Fall, helps recoup a little of what was lost, since man no longer had access to the Tree of Life after sin (Genesis 3:22).
In particular, I suspect sleep was a part of the original creation as a reminder of what God did during Creation Week. God, at the end of His work of creation, rested. Later, Moses presented the Israelites with The Ten Commandments, and in Exodus 20:11, God points out that the basis for God resting was a pattern for man to follow. So, man was created to rest—even before the Fall. Also, since God created night during Creation Week, it would be logical that Adam and Eve would have needed rest and sleep before the Fall.
Today, in our fallen world, proper amounts of sleep often refresh the mind and the body. While there’s no definitive proof Adam and Eve slept prior to the Fall, rest—in the form of sleep—seems to have been part of the original creation. But take the time to examine the issue yourself.