Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the resulting 3manifold have a hyperbolic structure with finite volume? I am tempted to say 'no' because it would have an end which is cylindrical (not toroidal), but know little enough about the subject to be uncertain.

An end of an orientable finite volume hyperbolic $3$manifold always has a neighborhood homeomorphic to $S^1 \times S^1 \times \mathbb{R}$, so no. Introductory texts on hyperbolic manifolds will contain this result. In your case, you could simply check that any neighborhood of the end has infinite volume. 

