This isn't a very general answer, but it is a convenient and significant one. You can read off the typical dimension of a random representation, by the hook length formula. Of course it is not as simple as, oh here's a formula, because you have to check whether the formula is stable. However, the hook-length formula is a factorial divided by a product of hook lengths. So you can check that the logarithm of the formula is indeed statistically stable. Up to normalization, it limits to a well-behaved integral over the Kerov-Vershik shape.

The dimension of a group representation is of course $\chi(1)$, the trace of the identity. Given the nice behavior of this statistic in a random representation, it is natural to ask about the typical value of $\chi(\sigma)$ for some other type of permutation $\sigma$. Two problems arise. First, $\sigma$ isn't really one type of permutation, but rather some natural infinite sequence of permutations. Second, the Murnaghan–Nakayama formula for $\chi(\sigma)$, and probably any fully general rule, isn't statistically stable. The Murnaghan–Nakayama rule is a recursive alternating sum; in order to apply it to a large Plancherel-random representation you would have to know a lot about the local statistics of its tableau, and not just its shape. For instance, suppose that $\sigma$ is a transposition. Then the MN rule tells you to take a certain alternating sum over rim dominos of the tableau $\lambda$. (The sign is positive for the horizontal dominos and negative for the vertical dominos.) I suspect that there is a typical value for $\chi(\sigma)$ when $\sigma$ is a transposition, or probably any permutation of fixed type that is local in the sense that a transposition is local. But this would use an elaborate refinement of the Kerov–Vershik theorem, analogous to the local central limit theorem augmented by a local difference operator, and not just the original Kerov–Vershik.

However, I did find another character limit in this spirit that is better behaved. In On the number of rim hook tableaux, Fomin and Lulov established a product formula for the number of $r$-rim hook tableaux, which is also $\chi(\sigma)$ when $\sigma$ is a "free" permutation consisting entirely of $r$-cycles (and no fixed points or cycle lengths that are factors of $r$). This includes the important case of fixed-point-free involutions. If $\sigma$ acts on $mr$ letters, then according to them, the number of these is
$$\chi_\lambda(\sigma) = \frac{m!}{\prod_{r|h(t)} (h(t)/r)},$$
where $h(t)$ is the hook length of the hook at some position $t$ in the shape $\lambda$.

Happily, this is just a product formula and not an alternating sum or even a positive sum. To approximate the logarithm of this character with an integral, you only need a mild refinement of Kerov–Vershik, one that says that the hook length $h(t)$ of a typical position $t$ is uniformly random modulo $r$. (So this is a good asymptotic argument when $r$ is fixed or only grows slowly.)

**Correction:** Per a comment, JSE already thought of the first part of my answer, which I stated overconfidently. The estimate for $\log \chi(1)$ (and in the other cases of course) is an improper integral, I guess, so it does not follow just from the statement of Kerov–Vershik that the integral gives you an accurate estimate of the form
$$\log \chi(1) = C\sqrt{n}(1+o(1)).$$
However, it looks like these issues have been swept away by later, stronger versions of the original Kerov–Vershik result. The arXiv paper Ivanov and Olshanki - Kerov's central limit theorem for the Plancherel measure on Young diagrams establishes not just a typical limit for the dimension (and other character values), but also a central limit theorem.