  We study the Whittaker category $\mathcal N(\zeta)$ of the Lie superalgebra
$\mathfrak g$ for an arbitrary character $\zeta$ of the even subalgebra of the
nilpotent radical associated with a triangular decomposition of $\mathfrak g$.
We prove that the Backelin functor from either the integral subcategory or any
strongly typical block of the BGG category to the Whittaker category sends
irreducible modules to irreducible modules or zero. The category $\mathcal
N(\zeta)$ provides a suitable framework for studying finite $W$-superalgebras
associated with an even principal nilpotent element. For the periplectic Lie
superalgebras $\mathfrak{p}(n)$, we formulate the principal finite
$W$-superalgebras $W_\zeta$ and establish a Skryabin-type equivalence. For a
basic classical and a strange Lie superalgebras, we prove that the category of
finite-dimensional modules over a given principal finite $W$-superalgebra
$W_\zeta$ is equivalent to $\mathcal N(\zeta)$ under the Skryabin equivalence,
for a non-singular character $\zeta$. As a consequence, we give a super
analogue of Soergel's Struktursatz for a certain Whittaker functor from the
integral BGG category $\mathcal O$ to the category of finite-dimensional
modules over $W_\zeta$.
