What's the Usefulness of Prognostic Sphinx's Second Ability?
Prognostic Sphinx's ability has no restriction on timing, so you can activate it whenever you have priority. This includes on your opponent's turn, or in response to another spell or ability. 

Abilities that have timing restrictions will always state it, either directly on the card or in the rules for that ability. For example, the rule for Outlast has a clause "Activate this ability only any time you could cast a sorcery." This clause restricts when you can activate the ability.

Creatures do not untap on your opponent's turn unless an effect specifically untaps them.

Tapping Prognostic Sphinx is not part of the cost of activating the ability. It is an effect. This is important because it means you can activate the ability even when Prognostic Sphinx is already tapped (although the instruction to "Tap it." won't actually do anything).