I have 3 islands and a Misty Rainforest in play. Two of the islands are tapped. My opponent casts a spell that I want to counter with the deprive I'm holding. I crack the Misty and grab a Mystic Sanctuary. When the Sanctuary comes into play, it's etb ability goes on the stack. Before that ability resolves, can I tap island+Sanctuary to cast deprive, bounce the Sanctuary to my hand (with Deprive's additional cost), and resolve the sanctuary's etb ability to put the now resolved deprive on top of my library?
When Mystic Sanctuary enters the battlefield in this scenario, its last ability triggers and goes on the stack before any player gets priority to cast spells, including Deprive (C.R. 117.3b, 117.5, 117.1a). And once that ability goes on the stack you must target an "instant or sorcery card [in] your graveyard", if possible, right then and there (C.R. 603.3d, 601.2c). (If there is no such card, the ability is removed from the stack [C.R. 603.3d, 601.2c].) Note that choosing the required number of targets for a spell or ability isn't optional, even if its effect is optional because it uses the word "may" (C.R. 603.5, 603.3d, 601.2c; compare Mystic Sanctuary with Sepulchral Primordial and see also this thread and this thread).
However, you can still cast Deprive in this scenario; as you cast Deprive, you can tap Mystic Sanctuary and another Island you control for UU in total (assuming both were untapped before you did so), then return Mystic Sanctuary to its owner's hand and pay UU — note that you activate mana abilities (including tapping lands you control for mana) before you pay the spell's costs (C.R. 601.2g-h).
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However, you can still cast Deprive in this scenario; as you cast Deprive, you can tap Mystic Sanctuary and another Island you control for UU in total (assuming both were untapped before you did so), then return Mystic Sanctuary to its owner's hand and pay UU — note that you activate mana abilities (including tapping lands you control for mana) before you pay the spell's costs (C.R. 601.2g-h).