This is how the wizard knows to cast shield. When the arrow does indeed strike it's target is when you roll damage.
You keep referring to a reaction situation that usually happens with the wizard knowing who is attacking and with what. In this situation, the wizard does not know who is attacking or with what, they don't even know there is an attack coming! At the very least, the wizard should have to succeed on an active perception roll prior to any reaction if the attacker is stealthy/hidden.
There is definitely a conflict in the rules here. On the one hand, there are those who think the rule saying those surprised cannot move or act on their turn and cannot react until after their turn means surprise ends at the end of the turn they could not move or act in. On the other hand, there are those who think the rule saying that a stealth/passive perception check determines whether anyone is surprised means the only things that will end surprise are a successful active perception roll or the target's realization that they have been shot at (because the arrow is either in their back or in the tree next to them). I don't see any way to reconcile these two views. The first view will never accept that anyone can still be surprised once they are allowed to react, and the second view will never accept that initiative, not stealth/perception, determines when surprise ends.
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