Tom's comment gives a generic answer that applies to $Top^\mathcal{J}$ for any small topological
category $\mathcal{J}$. For the orbit category $\mathcal{O}_G$, where $G$ is any discrete group
(not necessarily finite), something special and much nicer happens. We have the fixed point functor $\Phi$ from $G$-spaces to $\mathcal{O}_G$-spaces. It sends $X$ to the functor on the orbit category that sends $G/H$ to $X^H$. The functor $\Phi$ actually specifies an isomorphism from the category of $G$-cell complexes (or the category of $G$-CW complexes) to the category of cell diagrams or (CW diagrams) in the diagram category of $\mathcal{O}_G$-spaces. Modulo a few obvious typos, the easy proof is given on page 56 of Equivariant Homotopy and Cohomology Theory (by some of my friends and me),
CBMS Regional Conference Series in Mathematics Number 91, AMS 1996. That book is posted on my web page
(http://math.uchicago.edu/~may/). The cofibrations are the retracts of the relative cell diagrams.