In this paper, we characterize the extremal digraphs with the maximal or minimal $α$-spectral radius among some digraph classes such as rose digraphs, generalized theta digraphs and tri-ring digraphs with given size $m$. These digraph classes are denoted by $\mathcal{R}_{m}^k$, $\widetilde{\boldsymbolΘ}_k(m)$ and $\INF(m)$ respectively. The main results about spectral extremal digraph by Guo and Liu in \cite{MR2954483} and Li and Wang in \cite{MR3777498} are generalized to $α$-spectral graph theory. As a by-product of our main results, an open problem in \cite{MR3777498} is answered. Furthermore, we determine the digraphs with the first three minimal $α$-spectral radius among all strongly connected digraphs. Meanwhile, we determine the unique digraph with the fourth minimal $α$-spectral radius among all strongly connected digraphs for $0\le α\le \frac{1}{2}$.