One simple (but not very nice) idea that I can think of is processing the queries offline. Find the topological sort of the dag and traverse from reverse and maintain bitsets.

Since everyone is talking about bitsets, I want to note one thing — having $$$N$$$ bitsets of length $$$N$$$ will be more than 1 GB, which is more than the memory limit on most platforms (and I'm not sure how fast you can allocate it, either).

Instead, don't use bitsets, just long longs. Maintain dp[u], where dp[u] is a bitmask telling you which of the first 64 vertices are reachable from $$$u$$$. Then do the same thing for vertices 65...128 and so on. You'll make $$$\frac{N}{64}$$$ passes, each of them with time complexity $$$O(N)$$$, so you get the same complexity as the bitset solution without the memory implications.

(Note that this kills off online solutions...)