Simultaneously reflecting and transmitting reconfigurable intelligent surfaces (STAR-RIS) has recently emerged as prominent technology that exploits the transmissive property of RIS to mitigate the half-space coverage limitation of conventional RIS operating on millimeter-wave (mmWave). In this paper, we study a downlink STAR-RIS-based multi-user multiple-input single-output (MU-MISO) mmWave hybrid non-orthogonal multiple access (H-NOMA) wireless network, where a sum-rate maximization problem has been formulated. The design of active and passive beamforming vectors, time and power allocation for H-NOMA is a highly coupled non-convex problem. To handle the problem, we propose an optimization framework based on alternating optimization (AO) that iteratively solves active and passive beamforming sub-problems. Channel correlations and channel strength-based techniques have been proposed for a specific case of two-user optimal clustering and decoding order assignment, respectively, for which analytical solutions to joint power and time allocation for H-NOMA have also been derived. Simulation results show that: 1) the proposed framework leveraging H-NOMA outperforms conventional OMA and NOMA to maximize the achievable sum-rate; 2) using the proposed framework, the supported number of clusters for the given design constraints can be increased considerably; 3) through STAR-RIS, the number of elements can be significantly reduced as compared to conventional RIS to ensure a similar quality-of-service (QoS).