Shape memory alloys (SMAs) are functional materials with a wide range of applications, from the aerospace sector to the biomedical field. Nowadays, there is a worldwide interest in developing SMAs through powder metallurgy like additive manufacturing (AM), which allows innovative building processes. However, producing SMAs using AM techniques is particularly challenging because of the microstructure required to obtain optimal functional properties. This aspect is critical in the case of Cu–Al–based SMAs, due to their high elastic anisotropy, making them brittle in polycrystalline form. In this work, we approached the processing of a Cu–Al–Ni SMA following a specific powder metallurgy route: gas atomization of a pre-alloyed melt; compaction of the atomized powders through hot isostatic pressing; and a final hot rolling plus thermal treatments. Then, the microstructure of the material was characterized by electron microscopy showing a specific [001] texture in the rolling direction that improved the functional behavior. The successive processing steps produce an increase of about 40 °C in the martensitic transformation temperatures, which can be well controlled and reproduced through the developed methodology. The thermomechanical functional properties of superelasticity and shape memory were evaluated on the final SMA. Outstanding, fully recoverable superelastic behavior of 4.5% in tension, as well as a ±5% full shape memory recovery in bending, were reported for many cycles. These experiments demonstrate the enhanced mechanical and functional properties obtained in polycrystalline Cu–Al–Ni SMAs by powder metallurgy. The present results pave the road for producing this kind of SMA with the new AM technologies, which always produce polycrystalline components and can improve their processes taking the powder metallurgy SMA, here produced, as reference material.