Essentially, Sphinx of Jwar Isle lets you play with the top card of your library revealed only to you. Knowing what that card is becomes part of the information you have access to, just like you can look at the cards in your hand. You may look at the top card of your library whenever you want, even if you don't have priority. This action doesn't use the stack.
If the top card of your library changes during the process of casting a spell or activating an ability, you can't look at the new top card until the process of casting the spell or activating the ability ends (all targets are chosen, all costs are paid, and so on).