Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. A strong subgraph $H$ of $D$ is called an $S$-strong subgraph if $S\subseteq V(H)$. A pair of $S$-strong subgraphs $D_1$ and $D_2$ are said to be arc-disjoint if $A(D_1)\cap A(D_2)=\emptyset$. Let $\lambda_S(D)$ be the maximum number of pairwise arc-disjoint $S$-strong subgraphs in $D$. The strong subgraph $k$-arc-connectivity is defined as $$ \lambda_k(D)=\min\{\lambda_S(D)| S\subseteq V(D), |S|=k\}. $$ The parameter $\lambda_k(D)$ can be seen as a generalization of classical edge-connectivity of undirected graphs. In this paper, we first obtain a formula for the arc-connectivity of Cartesian product $\lambda(G\Box H)$ of two digraphs $G$ and $H$ generalizing a formula for edge-connectivity of Cartesian product of two undirected graphs obtained by Xu and Yang (2006). Using this formula, we get a new formula for the arc-connectivity of Cartesion product of $k\ge 2$ copies of a strong digraph $G$: $\lambda(G^{k})=k\cdot \min\left \{ \delta ^{+}\left ( G \right ),\delta ^{-} \left ( G \right )\right \}$. Then we study the strong subgraph 2-arc-connectivity of Cartesian product $\lambda_2(G\Box H)$ and prove that $ \min\left \{ \lambda (G) |H|, \lambda (H)|G|, %\\ \delta ^{+ }(G)+ \delta ^{+ } (H), \right.$ \linebreak $\left.\delta ^{-} (G)+ \delta ^{- } (H) \right \}\ge\lambda_2(G\Box H)\ge \lambda_2(G)+\lambda_2(H)-1.$ The upper bound for $\lambda_2(G\Box H)$ is sharp and is a simple corollary of the formula for $\lambda(G\Box H)$. The lower bound for $\lambda_2(G\Box H)$ is either sharp or almost sharp i.e., differs by 1 from the sharp bound. We improve the lower bound under an additional condition and prove its sharpness by showing that $\lambda_2(G\Box H)\geq \lambda_2(G)+ \lambda_2(H),$ where $G$ and $H$ are two strong digraphs such that $\delta ^{+}\left (H \right)> \lambda _{2}(H)$. We also obtain exact values for $\lambda_2(G\Box H)$, where $G$ and $H$ are digraphs from some digraph families.