NiTi-Cu shape memory alloys (SMAs) represent a critical class of functional materials renowned for their superior superelasticity, shape memory effect (SME), and tunable thermomechanical responses, which are essential for advanced applications in aerospace, biomedical systems, and smart actuators. The strategic incorporation of copper (Cu) into NiTi alloys serves as a powerful approach to modulate phase transformation temperatures, minimize thermal hysteresis, and enhance mechanical and elastocaloric stability. However, producing NiTi-Cu SMAs via fusion-based additive manufacturing (AM) methods remains challenging due to high thermal gradients, residual stress accumulation, and non-uniform phase distribution. Binder Jet Additive Manufacturing (BJAM), a powder sintering-based technique, has emerged as a promising non-fusion alternative capable of fabricating intricate geometries while mitigating thermal-induced defects. This review presents a comprehensive and critical evaluation of recent progress in the fabrication of NiTi-Cu SMAs, with a distinct focus on the potential and optimization of the BJAM process. Key topics include the influence of Cu on phase transformation pathways, microstructural evolution, and the elastocaloric performance of NiTi-Cu alloys. In addition, this review has examined the influence of particle morphology, binder chemistry, sintering dynamics, and post-processing strategies on densification, mechanical behavior, and shape memory functionality. Special attention is given to the integration of computational tools such as CALPHAD and machine learning (ML) for predictive alloy design and process optimization, offering a data-driven roadmap for scalable production. This review identifies critical knowledge gaps, including the lack of experimental studies on BJAM-specific NiTi-Cu fabrication and the challenges associated with Cu segregation, intermetallic formation, and contamination during sintering. By consolidating foundational insights and proposing a structured research framework, this study aims to advance the scientific understanding and industrial applicability of BJAM in the manufacturing of high-performance NiTi-Cu SMAs, ultimately bridging the gap between fundamental research and practical deployment in smart material systems.