The Max-DICUT problem has gained a lot of attention in the streaming setting in recent years, and has so far served as a canonical problem for designing algorithms for general constraint satisfaction problems (CSPs) in this setting. A seminal result of Kapralov and Krachun [STOC 2019] shows that it is impossible to beat $1/2$-approximation for Max-DICUT in sublinear space in the single-pass streaming setting, even on bounded-degree graphs. In a recent work, Saxena, Singer, Sudan, and Velusamy [SODA 2025] prove that the above lower bound is tight by giving a single-pass algorithm for bounded-degree graphs that achieves $(1/2-ε)$-approximation in sublinear space, for every constant $ε>0$. For arbitrary graphs of unbounded degree, they give an $O(1/ε)$-pass $O(\log n)$ space algorithm. Their work left open the question of obtaining $1/2$-approximation for arbitrary graphs in the single-pass setting in sublinear space. We make progress towards this question and give a two-pass algorithm that achieves $(1/2-ε)$-approximation in sublinear space, for every constant $ε>0$.