Thanks to Mark Wildon and the OP for pointing out that my answer was incorrect- I was considering the centralizer of $\sigma$ in the wrong algebra.
    However, it does seem to me that the structure of $A = C_{\mathbb{C}S_{N}}(\sigma)$ may depend on the prime factorization of $N.$

In general it is well-known that if $\lambda$ is a partition of $N$ and $\chi_{\lambda}$ is the associated complex irreducible character of $S_{N}$, then $\chi_{\lambda}(\sigma) \in \{0,1,-1 \}$.

When $N= p$ is prime, this implies that ${\rm Res}^{S_{N}}_{\langle \sigma \rangle}(\chi_{\lambda})$ has one of the forms $t\rho, t\rho + 1, t\rho -1$ where $t$ is a non-negative integer and $\rho$ is the regular characer of $\sigma$. But this inference can not be drawn when $N$ is not prime, and is indeed false in general (for example, when $N = 4$ and $\chi_{\lambda}$ has a Klein $4$-subgroup in its kernel). (to be continued...)