The mapping class groups of all compact orientable 3-manifolds are essentially known. A fairly detailed summary of the results, focusing on the nonprime case and with references to proofs in the literature, can be found in Section 2 of a paper of mine with Nathalie Wahl: "Stabilization for mapping class groups of 3-manifolds", Duke Math. J. 155 (2010), 205-269, arXiv:0709.2173. (Section 2 can be read independently of the rest of the paper.) The results are stated in terms of the map $\Phi:MCG(M)\to Out(\pi_1M)$. For $M$ closed and orientable the kernel of $\Phi$ is a direct sum of finitely many copies of ${\mathbb Z}_2$ generated by Dehn twists along 2-spheres. Also, $\Phi$ is surjective if this is true for each prime connected summand of $M$ and a certain other natural condition holds involving permuting different summands with isomorphic fundamental groups.
In the case of $T^3 \# (S^1\times S^2)$ the kernel of $\Phi$ is a single ${\mathbb Z}_2$ generated by a Dehn twist along the $S^2$ factor of $S^1\times S^2$. (Interestingly, the twist along the $S^2$ separating the two connected summands is isotopic to the identity.) The image of $\Phi$ is an index two subgroup of $Out({\mathbb Z}^3*{\mathbb Z})$ since one can only realize $SL(3,{\mathbb Z})$, not all of $GL(3,{\mathbb Z})$, by orientation-preserving homeomorphisms of $T^3$.
The structure of the automorphism group of a free product seems to be well understood in terms of the automorphism groups of the factors. For $M=T^3 \# (S^1\times S^2)$ this means its mapping class group is generated by the MCG's $SL(3,{\mathbb Z})$ and ${\mathbb Z}_2 \times {\mathbb Z}_2$ of the two summands, along with homeomorphisms obtained by regarding $M$ as obtained from $T^3$ by attaching a "1-handle" and dragging either end of this handle around loops in $T^3$.