We study D-branes in the Nappi-Witten model, which is a gauged WZW model based on (SL(2, R)×SU (2))/(U (1)×U (1)). The model describes a four dimensional space-time consisting of cosmological regions with big bang/big crunch singularities and static regions with closed time-like curves. The aim of this paper is to investigate by D-brane probes whether there are pathologies associated with the cosmological singularities and the closed time-like curves. We first classify D-branes in a group theoretical way, and then examine DBI actions for effective theories on the D-branes.In particular, we show that D-brane metric from the DBI action does not include singularities, and wave functions on the D-branes are well behaved even in the presence of closed time-like curves. * The formulation of string theory in cosmological or broadly time-dependent backgrounds still remains as one of the most exciting open challenges for theoretical physicists. Among other things, it is important to see how (whether) string theory resolves the cosmological singularities (or not). Some recent attempts along this direction have been made in [1,2,3,4,5,6,7,8,9,10,11,12,13]. In this paper, we investigate the Nappi-Witten (NW) model [14,15] as a conformal field theory (CFT) model that includes big bang and big crunch singularities. This model is defined as an SL(2, R) × SU(2) WZW model gauged by U(1)×U(1) in an asymmetric way. Therefore we can solve it exactly and gain some understanding of the behaviour of strings across these singularities. Similar attempts have been made to understand the cosmological backgrounds by using gauged WZW models, e.g., in [16,17,18,19,20].The NW model describes a four dimensional anisotropic universe that starts at a big bang singularity, expands for a while, ends at a big crunch singularity within a finite affine time, and again starts at another big bang singularity. It was argued in [15] that strings across these singularities make sense because the theory is defined as a coset CFT.This may be related to the fact that strings have a finite length scale l s and may go through the singularities. As described in [15], the coset contains, in addition to copies of the cosmological regions, non-compact and time independent (static) regions known as whiskers. The whisker part of the full space-time contains closed time-like curves (CTCs), which seems to be unacceptable for any measurable quantity like S-matrix. Nevertheless, these CTCs are argued to be necessary for defining the observables in terms of gauged WZW model [15].Hence we would like to ask a question, whether studying the dynamics of D-branes sheds more lights on the understanding of big bang/big crunch singularities and CTCs. This is a non-trivial question because the D-branes, or the open strings ending on them, feel the background geometry in a different way. In particular, D0-brane is an useful candidate to probe the singularities, because the point particle could probe directly the singular points. In fact, the singularities are at strong coupling reg...