This article is concerned with an inverse problem of simultaneously determining a spatially varying coefficient and a Robin coefficient for a one-dimensional fractional diffusion equation. The equation incorporates a time-fractional derivative of order alpha is an element of (0,1) and non-homogeneous boundary conditions. We prove the uniqueness for the inverse problem by observation data at one interior point over a finite-time interval, provided that a coefficient is known on a subinterval. Our proof is based on the uniqueness in the inverse spectral problem for a Sturm-Liouville problem by means of the Weyl mfunction and the spectral representation of the solution to an initial-boundary value problem for the fractional diffusion equation.