  We theoretically investigate optical (frequency-dependent) bulk spin
transport properties in a spin-1/2 topological Fermi superfluid. We
specifically consider a one-dimensional system with an interspin {\it p}-wave
interaction, which can be realized in ultracold atom experiments. Developing
the BCS-Leggett theory to describe the Bardeen-Cooper-Schrieffer (BCS) to
Bose-Einstein condensate (BEC) evolution and the $\mathbb{Z}_2$ topological
phase transition in this system, we show how the spin transport reflects these
many-body aspects. We find that the optical spin conductivity, which is a small
AC response of a spin current, shows the spin gapped spectrum in the wide
parameter region and the gap closes at $\mathbb{Z}_2$ topological phase
transition point. Moreover, the validity of the low-energy effective model of
the Majorana zero mode is discussed along the BCS-BEC evolution in connection
with the scale invariance at {\it p}-wave unitarity.
