In this paper, we define generalized Bishop frame of a null Cartan curve in Minkowski $3$-space by using its Bishop's frame vector fields. We obtain the Cartan equations according to the generalized Bishop frame and give the relations between the generalized Bishop curvatures and Bishop curvatures. We show that among all null Cartan curves in $\mathbb{E}^{3}_{1}$, only the null Cartan cubic has two generalized Bishop frames, one of which coincides with its Bishop frame. We also show that there exists a null Cartan curve whose generalized Bishop curvatures and Bishop curvatures are equal, but whose generalized Bishop frame and Bishop frame do not coincide. As an application, we characterize a $k$-type null Cartan slant helices for $k\in\{0,1,2\}$ according to their generalized Bishop frame and give some examples.