This paper proposes a simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) aided wireless-powered non-orthogonal multiple access (NOMA) system, which includes an access point (AP), a STAR-RIS, and two non-orthogonal users located at both sides of the STAR-RIS. In this system, the users first harvest the radio-frequency energy from the AP in the downlink, then adopt the harvested energy to transmit information to the AP in the uplink concurrently. Two policies are considered for the proposed system. The first one assumes that the time-switching protocol is used in the downlink while the energy-splitting protocol is adopted in the uplink, named TEP. The second one assumes that the energy-splitting protocol is utilized in both the downlink and uplink, named EEP. The outage probability, sum throughput, and average age of information (AoI) of the proposed system with TEP and EEP are investigated over Nakagami-m fading channels. In addition, we also analyze the outage probability, sum throughput, and average AoI of the STAR-RIS aided wireless-powered time-division-multiple-access (TDMA) system. Simulation and numerical results show that the proposed system with TEP and EEP outperforms baseline schemes, and significantly improves sum throughput performance but reduces outage probability and average AoI performance compared to the STAR-RIS aided wireless-powered TDMA system. Furthermore, to maximize the sum throughput and ensure a certain average AoI, we design a genetic-algorithm based time allocation and power allocation (GA-TAPA) algorithm. Simulation results demonstrate that the proposed GA-TAPA method can significantly improve the sum throughput by adaptively adjusting system parameters.