I'm unsure if this would be termed a 'simple rule' but perhaps this will be of some use: the *Kostka-Foulkes polynomials* $K_{\lambda\mu}(q)$ ($\lambda,\mu$ partitions of $n$) determine the change of basis matrix in $\Lambda(q)$, the one variable ring of symmetric functions, between the Hall-Littlewood polynomials and the Schur polynomials (described in Macdonald's 'Symmetric Functions and Hall Polynomials', for example). Garsia-Procesi proved (Adv. in Math, 94, p.82-138) that the Kostka-Foulkes polynomials determine the graded decomposition of the cohomology of of Springer fibres

$$X_{\lambda} = \{ (F_{\cdot})\in Fl(n)\;|\; U_{\lambda}(F_{i})\subset F_{i-1}\}$$

where $U_{\lambda}$ is a nilpotent matrix of Jordan type $\lambda$: they show that

$$K_{\mu\lambda}(q) = q^{n(\lambda)}\sum_{i} \langle H^{2i}(X_{\lambda},\mathbb{Q}), \chi^{\mu}\rangle q^{-i}.$$

Here $\chi^{\mu}$ is the character of the Specht module associated to the partition $\mu$, $\langle P, Q\rangle$ is the multiplicity of an irreducible $S_{n}$-module $Q$ appearing in an $S_{n}$-module $P$ and $n(\lambda)$ is the (complex) dimension of $X_{\lambda}$ (see Garsia-Procesi, formula (I.9)).

In earlier work, Lascoux and Schutzenberger proved that the Kostka-Foulkes polynomials $K_{\mu\lambda}(q)$ can be determined using the *charge statistic* $c$: this associates to a tableaux of shape $\mu$ and weight $\lambda$ some non-negative integer - this is described in Macdonald's book, p.242. They showed that

$$K_{\mu\lambda}(q) = \sum_{T\in SSYT(\lambda;\mu)} q^{c(T)}$$

where $SSYT(\lambda;\mu)$ is the set of semistandard Young tableau of shape $\mu$ and weight $\lambda$

In your case, we have $\lambda=(1^{n})$, so that the charge statistic associates to a standard tableau $T$ some integer $c(T)$. Thus, to determine the multiplicity of $\chi^{\mu}$ in $H^{2i}(Fl(n))$ you can determine the coefficient of $q^{n(n-1)/2-i}$ in $K_{\mu(1^{n})}(q)$ using the charge statistic; in short, the multiplicity of $\chi^{\mu}$ in $H^{2i}(Fl(n))$ equals the number of standard tableau $T$ of shape $\mu$ with charge $c(T)=n(n-1)/2 -i$.