In this article, Intelligent Reflecting Surface (IRS) technology is used to assist a Wireless Powered Communication Network (WPCN) system composed of a Hybrid Access Point (HAP) and multiple devices during downlink and uplink phases. In the downlink, devices charge their batteries through Wireless Power Transfer (WPT) from the power signals coming directly from the HAP and from the signals reflected by the IRS. In the uplink, the devices transmit their information signals to the HAP by employing Non-Orthogonal Multiple Access (NOMA). In the HAP, information is decoded through Successive Interference Cancellation (SIC). Based on this system model, an optimization problem is formulated so as to maximize the total uplink data rate subject to individual Quality of Service (QoS) guarantees for devices where IRS’s phase shifts, uplink/downlink time duration and device ordering in SIC are optimized. Motivated by the high complexity of the optimization problem, we employ alternate optimization where Genetic Algorithm (GA) is used to define IRS’s phase shifts in uplink/downlink and a heuristic solution is employed to optimize the remaining variables. Simulation results show that the proposed solution is able to satisfy different levels of QoS demands with higher spectral efficiency compared to baseline solutions.