In this paper, we study the holomorphic de Rham cohomology of a compact strongly pseudoconvex CR manifold $X$ in $\mathbb{C}^N$ with a transversal holomorphic $S^1$-action. The holomorphic de Rham cohomology is derived from the Kohn-Rossi cohomology and is particularly interesting when $X$ is of real dimension three and the Kohn-Rossi cohomology is infinite dimensional. In Theorem A, we relate the holomorphic de Rham cohomology $H^k_h(X)$ to the punctured local holomorphic de Rham cohomology at the singularity in the variety $V$ which $X$ bounds. In case $X$ is of real codimension three in $\mathbb{C}^{n+1}$, we prove that $H^{n-1}_h(X)$ and $H^n_h(X)$ have the same dimension while all other $H^k_h(X)$, $k>0$, vanish (Theorem B). If $X$ is three-dimensional and $V$ has at most rational singularities, we prove that $H^1_h(X)$ and $H^2_h(X)$ vanish (Theorem C). In case $X$ is three-dimensional and $N=3$, we obtain in Theorem D a complete characterization of the vanishing of the holomorphic de Rham cohomology of $X$.