Does a spell with cascade let you cast the revealed card for free even if you don't have priority when cascade triggers?

Cascade is a triggered ability (CR 702.84a). When you cast a spell with cascade, the cascade ability triggers and is placed on the stack. Like any triggered ability, it waits until a player would receive priority, then it is placed on the stack at that time (CR 603.3). The ability does not resolve — and you cannot cast the revealed card — until the ability reaches the top of the stack and resolves.

When the cascade trigger resolves, you exile cards from the top of your library until you exile a nonland card with lesser mana value, then you may cast that card without paying its mana cost (CR 702.84b). The casting happens as part of the trigger's resolution, not as a separate priority window you need to hold. So you do need priority in the sense that the trigger must resolve normally — but you don't separately need to 'have priority' to cast the revealed card, because casting it is part of resolution itself.

Importantly, opponents can respond to the cascade trigger before it resolves (e.g., by countering it), which would prevent you from casting the revealed card at all. Once the trigger is resolving, however, no one receives priority mid-resolution to interfere with the cast itself (CR 608.2).

Concrete example: You cast Shardless Agent (cascade). The cascade ability triggers and goes on the stack. Your opponent passes priority, you pass priority, and the trigger resolves. During resolution you exile cards until you hit a nonland card with mana value 2 or less — say, Brainstorm — and cast it for free right then. You did not need a separate priority step to cast Brainstorm; it happened inside the trigger's resolution.

So the short answer is: you don't need to 'have priority' as a separate action to cast the cascade card, but the trigger itself must resolve normally through the stack, which requires the normal priority process (CR 116.3, 702.84b).