Both non-orthogonal multiple access (NOMA), which can serve multiple users simultaneously and on the same frequency, and cognitive radio (CR) can contribute to eliminating the spectrum scarcity problem. In this work, an uplink CR-based NOMA (CR-NOMA) system, which is equipped with multiple users and a base station with a multi-antenna, is proposed to improve spectral efficiency. By considering the users' quality of service (QoS), the system performance of successive interference cancellation (SIC) is investigated in this system. Two different antenna and secondary user selection algorithms are proposed to improve the outage performance and retard the effect of the error floor. The multi-antenna CR-NOMA with QoS-based SIC system outperforms conventional channel state information (CSI)-based SIC. In addition, it is shown that the outage performance of this system on both proposed algorithms is better than in the case of not using the algorithm. The closed-form outage probability expression of this system for the suboptimal joint antenna and user selection algorithm is derived. Furthermore, when the proposed algorithms are not used, the closed-form expression of the outage probability for this system with a single-antenna base station is derived. Extensive simulation results verify the accuracy of theoretical analyses.