The expression "watching unto prayer" means missing out on sleep rather than going without prayer. Like a fast from food this requires forgoing another natural appetite--the need for sleep.
When He walked this earth, Jesus would go without sleep from time to time. Sometimes He would get up very early in the morning, other times He would stay up late. The night before He chose the twelve disciples, He didn't sleep at all due to His prayer activity. Maybe the reason He could sleep through a storm was an all-night prayer vigil.
When Paul and Silas were beaten and imprisoned for healing a demon-possessed slave girl, they couldn't sleep, so they prayed and sang praise songs. Of course, if we were flogged and put in stocks we probably wouldn't sleep much either. What was significant was that they turned their inability to sleep into a time of prayer and worship. As a result, they saved the Philippian jailer and his entire family.