The triggered ability triggers both when Frenzied Fugue enters the battlefield and at the beginning of your upkeep. You don't have to choose only one.

You won't control the enchanted permanent as your upkeep begins. Any "at the beginning of your upkeep" abilities it has won't trigger during your upkeep (unless you already controlled the permanent). Similarly, if you have an "at the beginning of your upkeep" ability that targets a permanent you control, that ability can't target it.