In recent years, memristive crossbar-based neuromorphic computing systems (NCS) have provided a promising solution for neural network acceleration. However, stuck-at faults (SAFs) in the memristor devices significantly degrade the computing accuracy of NCS. Besides, memristors suffer from process variations, causing the deviation of actual programmed conductance from its target value. In this paper, we propose a unified robust network training framework for a memristive crossbar-based NCS, simultaneously taking the impacts of SAFs and variations into account. In order to incorporate SAFs and variations into the training process, an effective sampling strategy for SAF and an efficient variation injection technique based on the local reparameterization method are developed. Experimental results clearly demonstrate that the proposed training framework can boost the computation accuracy of NCS and improve the NCS robustness.