Spiking Neural Networks (SNNs) are promising biologically plausible models of computation which utilize a spiking binary activation function similar to that of biological neurons. SNNs are well positioned to process spatiotemporal data, and are advantageous in ultra-low power and real-time processing. Despite a large body of work on conventional artificial neural network accelerators, much less attention has been given to efficient SNN hardware accelerator design. In particular, SNNs exhibit inherent unstructured spatial and temporal firing sparsity, an opportunity yet to be fully explored for great hardware processing efficiency. In this work, we propose a novel systolic-array SNN accelerator architecture, called SpikeX, to take on the challenges and opportunities stemming from unstructured sparsity while taking into account the unique characteristics of spike-based computation. By developing an efficient dataflow targeting expensive multi-bit weight data movements, SpikeX reduces memory access and increases data sharing and hardware utilization for computations spanning across both time and space, thereby significantly improving energy efficiency and inference latency. Furthermore, recognizing the importance of SNN network and hardware co-design, we develop a co-optimization methodology facilitating not only hardware-aware SNN training but also hardware accelerator architecture search, allowing joint network weight parameter optimization and accelerator architectural reconfiguration. This end-to-end network/accelerator co-design approach offers a significant reduction of 15.1x-150.87x in energy-delay-product(EDP) without comprising model accuracy.