In this paper, we propose a simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) and energy buffer aided multiple-input single-output (MISO) simultaneous wireless information and power transfer (SWIPT) non-orthogonal multiple access (NOMA) system, which consists of a STAR-RIS, an access point (AP), and reflection users and transmission users with energy buffers. In the proposed system, the multi-antenna AP can transmit information and energy to several single-antenna reflection and transmission users simultaneously by the NOMA fashion in the downlink, where the power transfer and information transmission states of the users are modeled using Markov chains. The reflection and transmission users harvest and store the energy in energy buffers as additional power supplies, which are partially utilized for uplink information transmission. The power outage probability, information outage probability, sum throughput, and joint outage probability closed-form expressions of the proposed system are derived over Nakagami-m fading channels, which are validated via simulations. Results demonstrate that the proposed system achieves better performance as compared to the proposed system with discrete phase shifts, the STAR-RIS aided MISO SWIPT-NOMA buffer-less, conventional RIS and energy buffer aided MISO SWIPT-NOMA, and STAR-RIS and energy buffer aided MISO SWIPT-time-division multiple access (TDMA) systems. Furthermore, a particle swarm optimization-based power allocation (PSO-PA) algorithm is designed to maximize the uplink sum throughput with a constraint on the uplink joint outage probability and Jain's fairness index (JFI). Simulation results illustrate that the proposed PSO-PA algorithm can achieve an improved sum throughput performance of the proposed system.