During a nap is correct. In a nap is not. But you can't say this sentence at all. It makes no sense It is impossible to fall asleep during a nap because a nap is a period of sleep. He can't fall asleep because he is already asleep.

It is unusual for him to fall asleep during nap-time. Napping means sleeping, so you can't say it's unusual to sleep during a nap. But you can talk about that period of time set aside for the nap, also known as nap-time.