It's not exact in general. Take $\mathcal{A}$ to be the category of finitely generated abelian groups, so that $\underline{\mathcal{A}}$ is the category of finite abelian groups and $\Omega=0$. Take the short exact sequence
$$0\rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2$$
and $A=\mathbb{Z}/2$. Then the long sequence is
$$\mathbb{Z}/2\stackrel{1}\rightarrow \mathbb{Z}/2\stackrel{0}\rightarrow \mathbb{Z}/2\rightarrow 0\rightarrow \cdots,$$
which is not exact.