Integrated sensing and communication (ISAC) system has been expected to play a vital role in future wireless networks and services. In this paper, we investigate a non-orthogonal multiple access (NOMA)-aided ISAC system in which the ISAC base station utilizes NOMA to serve multiple NOMA-users while performing radar sensing towards a group of sensing targets by leveraging the superimposed NOMA signals for NOMA-users. To investigate this problem, we formulate a joint optimization of the beamforming, the NOMA transmission duration and the sensing scheduling of sensing targets, with the objective of maximizing the sensing efficiency (i.e., the number of sensed targets within a time-interval) of ISAC system, while satisfying both the communication quality requirement of each NOMA-user and the sensing quality requirement of each selected targets. Despite that the formulated joint optimization problem is strictly non-convex, we propose a decomposition-based algorithm for solving the problem. Specifically, we exploit the successive convex approximation and penalty function method to equivalently transform the problem of optimizing the beamforming into a convex problem which can be addressed efficiently. Furthermore, with the optimal beamforming, we leverage the monotonic feature and adopt a bisection-search method to determine the NOMA transmission duration. Finally, we utilize an algorithm based on the cross-entropy learning to compute the optimal sensing scheduling. Numerical results verify the effectiveness of our proposed algorithms and show the performance advantage of our NOMA-aided ISAC system. Compared with some benchmark schemes, our NOMA-aided ISAC sensing scheduling scheme can achieve better performance in communication and sensing.