Then the answer is yes, finite dimensional irreducible $G_0$-modules are projective in the category of finite dimensional modules. 

Note that $G_0=G_0'\oplus Z$, where $G_0'$ is semisimple and $Z$ is the center. Let $V$ be a finite dimensional $G_0$-module. Then, by Weyl's theorem, $V$ is completely irreducible as a $G_0'$-module. The elements of $Z$ act as scalars on each irreducible summand of $V$, by Schur's lemma. This means that the decomposition of $V$ into irreducible $G_0'$-modules is a decomposition as $G_0$-modules. It now follows that the category of finite dimensional $G_0$-modules is semi-simple, and irreducible=projective.