Random access (RA) is a common technique to admit users to a network. Non-orthogonal multiple access-based RA (NOMA-RA) is a promising solution to support a large number of devices competing to access a limited number of radio resources. This paper aims to propose an intelligent access control and user scheduling technique for NOMA-RA by leveraging machine learning (ML) algorithms. We first theoretically derive the maximum throughput of NOMA-RA and the optimal access probabilities for all NOMA power levels, which can serve as the upper bound in the ideal environment. We then introduce our ML design based on multi-armed bandit (MAB) that controls users participation and their NOMA channel access to achieve the optimal throughput. Our ML design consists of two ML agents where the first agent manages the flow of traffic entering the preamble selection process and the second agent controls the user access to NOMA channels. To achieve the joint optimization of both decisions, the outcome of the first agent is used as a context for the second agent to synchronize its learning, while the overall performance is used as a feedback to both agents. Simulation experiments confirm the effectiveness of our joint agent design and its ability to make joint decisions to achieve the optimal performance.