We give a cyclic sieving phenomenon for symplectic $λ$-tableaux $SP(λ,2m)$, where $λ$ is a partition of an odd integer $n$ and $gcd(m,p)=1$ for any odd prime $p\leq n$. We use the crystal structure on Kashiwara-Nakashima symplectic tableaux to get a cyclic sieving action as the product $σ$ of simple reflections in the Weyl group. The cyclic sieving polynomial is the $q$-anologue of the hook-content formula for symplectic tableaux. More generally, we give a CSP for symplectic skew tableaux with analogous conditions on the shape and a cyclic group action that rotates tableaux weights in a way motivated by the $σ$-action.