No, there is a theorem of Deligne that the universal central extension of $Sp(2g,\mathbb{Z})$ is not residually finiteis not residually finite. If one takes a finite central extension (which is also not residually finite), then the elements in the center have finite conjugacy class, and all other elements have infinite conjugacy class, but there is no finite-index subgroup which has only one finite conjugacy class, since the center lies in the kernel of any homomorphism to a finite group.
It might be reasonable to consider your question for the class of residually finite groups.
