The answer is yes. The Jordan block decomposition of the generic nilpotent in $SL_m$ on a representation is the same as the decomposition of any representation under the principal $SL_2$ (which is a map of $SL_2$ to $SL_n$ which sends the generic nilpotent in $SL_2$ to a generic one in $SL_n$; people usually have a particular one in mind, but they are all the same up to conjugation by Jacobson-Morozov).

This can be extracted from the formula for the character of the principal $SL_2$ usually called the "quantum Weyl dimension formula"

$$\chi(V_\lambda)=\frac{\prod_{\alpha\in \Delta^+}q^{\langle\rho+\lambda,\alpha\rangle}-q^{-\langle\rho+\lambda,\alpha\rangle}}{\prod_{\alpha\in \Delta^+}q^{\langle\rho,\alpha\rangle}-q^{-\langle\rho,\alpha\rangle}}$$

One "only" needs to expand this out in terms of the characters of the irreducible $SL_2$ reps $\frac{q^{n+1}-q^{-n-1}}{q-q^{-1}}$.