No.  Let $X = E \times E$ with $E$ an elliptic curve and let $G = \mathbb Z_2 \oplus \mathbb Z_2$, with each factor acting on one of the $E$'s by the involution and fixing the other.  The quotient is $\mathbb P^1 \times \mathbb P^1$.  The effective cone of $E \times E$ is round, while the effective cone of $\mathbb P^1 \times \mathbb P^1$ is polyhedral.

Offhand I don't see an example where $Y$ has infinitely many rays and $X$ doesn't, but it wouldn't surprise me. It's true $X$ is a Mori dream space (which implies polyhedral effective cone), then so is $Y$.