Property of negligible functions
The function $n^{-\log(n)}=2^{-\log(n)^2}$ is negligible, but $n(\kappa)$ for it is $2^{\sqrt{\kappa}}$ which is not polynomial in $\kappa$.

If we look at the functions $\mu_c(n)=n^c$, their $n_c(\kappa)$ is $2^\frac{\kappa}{c}$, so if $\mu$ is negligible, its $n(\kappa)$ must be smaller than those (at least for big enough $\kappa$, this is ok since you asked for the asymptotics). This means that $n(\kappa)$ is subexponential, i.e., $n(\kappa)=2^{o(\kappa)}$.