I doubt that the answer here depends on whether or not the Orne Library is revealed before you use Duke's Investigate ability (if you start in another location).
When you initiate Duke's ability, first you pay costs (spend an action and exhaust Duke), then you resolve effects: move to a connecting location, then investigate your current location. Whether Orne Library was revealed or not should be irrelevant here.
Basically, the cost of the action changed during the resolution of that action's effect. Although it might be possible that you must spend an extra action during the resolution of the effect, and if you don't you'd end up moving to the Library but not investigating. As with Rougarou + Taunt, it's kinda hard to tell *when* you pay the additional cost.
I suggest you ask Matt about his interaction.
EDIT: Another possibility is that you make all player choices when you first initiate the ability, before you pay costs. If so, then you'd initiate the ability, declare that you want to move to a connecting location, then pay costs, then move and investigate. In this case it might indeed be relevant whether Orne Library is revealed or not.