Narcolepsy doesn’t keep the enchanted creature continually tapped. The creature untaps during its controller’s untap step as normal, then is tapped when Narcolepsy’s ability resolves during that upkeep. The creature may be tapped in response (for example, if it has an activated ability with {T} in its cost).

Narcolepsy’s triggered ability has an “intervening ‘if’ clause.” That means (1) the ability triggers only if the enchanted creature is untapped as any player’s upkeep begins, and (2) the ability does nothing if the enchanted creature is already tapped by the time it resolves.