I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known?

The only thing I know is this may be trivial thing : Fix a group element $g \in S_n$. Let $R(g)$ be the matrix of $g$ in the regular representation. If for an irrep say $\pi$ of $S_n$ if $\pi(g)$ (the matrix of $g$ in the irrep $\pi$) has an eigenvalue $\lambda$ with multiplicity $m_{\pi(g)}(\lambda)$ and $\pi$ has dimension $d_\pi$ then $\lambda$ will occur as an eigenvalue of the $R(g)$ with multiplicity at least $d_\pi m_{\pi(g) }(\lambda )$

It would be great if this same question can be answered for any other arbitrary representation of $S_n$.