Assuming stress-energy tensor ${\rm{T}}_{\mu}^{\nu}\equiv {\rm{diag}}~\{\varepsilon,-p,-p,-p\}$ and taking trace from both sides of Einstein field equations (EFE) one obtains the relation 
\begin{equation}
-R=\kappa~(\varepsilon-3 p). \tag{1}
\end{equation}
If you know your metric you can easily calculate the energy density and pressure from (in your notation) 
\begin{equation}\label{pressure}
\kappa~ p=-\frac{1-{A}^{-1}}{r^2}~,\tag{2}
\end{equation}
\begin{equation}
\label{density}
\kappa~\varepsilon=\frac{1-{A}^{-1}}{r^2}-\frac{1}{r}~\frac{{\rm d}{A}^{-1}}{{\rm d}r}~.\tag{3}
\end{equation}