I'm not sure if this precise formulation is standard linear algebra, but it is true. The important point is that $S$ acts locally finitely on $V$ (be careful, this only works because $S$ is commutative): if $v$ is a random vector, and $x_i$'s generators of $S$, then there's some minimal $m_i$ such that $x_i^{m_i}v=a_{m_i-1}x_i^{m_i-1}v+\cdots$, and the space $S\cdot v$ is spanned by monomials in the $x_i$'s where the power of $x_i$ is less than $m_i$ (just check that any linear combination of these times an $x_i$ can written in this form, using the relation above).
Thus, one can decompose any vector $v\in V$ by simply considering $S\cdot v$ and decomposing this using the finite dimensional result. This shows that $V$ is the sum of these subspaces, and their intersections are trivial essentially by definition.
For category $\mathcal{O}$ this is really unnecessary though; you can consider the action of the center on the endomorphism space (which is finite dimensional) of your module, and the projections of the identity to the different generalized eigenspaces will be idempotents projecting to the desired block decomposition.