Your question ("is $M$ homeomorphic to $T_f$?") is answered in the affirmative by Theorem 2 of Stallings' paper *On fibering certain 3-manifolds".  You will also need his Theorem 1.  Here are the statements (slightly simplified).  

**Theorem 1**: Suppose that $M$ is a compact connected three-manifold.  Suppose that $\Gamma$ is a finitely generated normal subgroup of $\pi_1(M)$ whose quotient group is $\mathbb{Z}$.  Then there is a surface $F$ properly embedded in $M$ so that $\Gamma = \pi_1(F)$. 

**Theorem 2**: With hypotheses as in Theorem 1.  Suppose that $M$ is irreducible.  Suppose that $\Gamma$ is not $\mathbb{Z}/2\mathbb{Z}$. Then $M$ is a surface bundle over the circle, with $F$ isotopic to a fibre.

That is, your hypothesis on the short exact sequence (plus Theorem 1) gives the surface $F$.  Your hypothesis that the manifold $M$ is hyperbolic then gives the additional hypotheses of Theorem 2. 

Note that Stallings does not cite Waldhausen. I suppose that this is because his situation is a very very simple case of a Haken hierarchy.  Once you have $F$ in your hands (and all the group theory hypotheses), it is "easy" to show that $M - F$ is homeomorphic to a product.