In this paper, a novel approach towards optimizing users' rate allocation and price customization in a nonorthogonal multiple access (NOMA) wireless network under quality of service (QoS)-differentiated requested services is proposed and studied. A multi-service wireless system is considered, where each user's QoS requirements are reflected through a utility function, alongside his willingness to pay for the corresponding service. Within this setting, in order to jointly allocate the customized price and rate, a two-variable optimization problem arises. Based on the principles of S-modular theory, the above two-variable (rate and price) optimization problem is modeled and solved as a distributed non-cooperative game. The existence and convergence to the Nash equilibrium point with reference to both user's uplink transmission rate and price is proven. The proposed approach, allowing for decision-making at the user side, is well aligned with the self-optimization and self-adaptation objectives of future emerging 5G wireless networks. The performance evaluation of the devised framework is conducted via modeling and simulation under various scenarios, and the numerical results clearly demonstrate its superiority against other existing approaches.