  In this paper, $N(\kappa)$-contact metric manifolds satisfying the conditions
$\widetilde{C}(\xi,X)\cdot\widetilde{C}=0$, $\widetilde{C}(\xi,X)\cdot R=0$,
$\widetilde{C}(\xi,X)\cdot S=0$, $\widetilde{C}(\xi,X)\cdot C=0$, $C\cdot S=0$
and $R\cdot C=f_{C}Q(g,C)$ have been investigated and obtained their
classification. Among others it is shown that a Weyl-pseudosymmetric
$N(\kappa)$-contact metric manifold is either locally isometric to the
Riemannian product $E^{n+1}(0)\times S^{n}(4)$ or an $\eta$-Einstein manifold.
Finally, an example is given.
