I know that a sequence of modules is exact iff the localization at each prime ideal is exact
What happens in the case we are working with graded modules? Can we say that a sequence is exact iff the localization at each homogeneous prime ideal is exact?
The answer is yes and the reason is that the functor $-\otimes R_{\mathfrak{m}}$ is faithfully exact in the category of graded $R$-modules; this is Proposition 1.5.15(c) in Bruns and Herzog.
In the above notation $R$ is our graded ring and it is assumed that it has a unique homogeneous ideal $\mathfrak{m}$, which is maximal with respect to all other homogeneous ideals, i.e., $R$ is $^*$local in the terminology of Bruns and Herzog.