Non-orthogonal multiple access (NOMA)-inspired integrated sensing and communication (ISAC) facilitates spectrum sharing for radar sensing and NOMA communications, whereas facing privacy and security challenges due to open wireless propagation. In this paper, active reconfigurable intelligent surface (RIS) is employed to aid covert communications in NOMA-inspired ISAC wireless system with the aim of maximizing the covert rate. Specifically, a dual-function base-station (BS) transmits the superposition signal to sense multiple targets, while achieving covert and reliable communications for a pair of NOMA covert and public users, respectively, in the presence of a warden. Two superposition transmission schemes, namely, the transmissions with dedicated sensing signal (w-DSS) and without dedicated sensing signal (w/o-DSS), are respectively considered in the formulations of the joint transmission and reflection beamforming optimization problems. Numerical results demonstrate that active-RIS-aided NOMA-ISAC system outperforms the passive-RIS-aided and without-RIS counterparts in terms of covert rate and trade-off between covert communication and sensing performance metrics. Finally, the w/o-DSS scheme, which omits the dedicated sensing signal, achieves a higher covert rate than the w-DSS scheme by allocating more transmit power for the covert transmissions, while preserving a comparable multi-target sensing performance.