In a content-delivery network, files' popularity and users' requests change fast. Conventional caching schemes, e.g., caching (re)placement once per day during the off-peak hours, may not capture the up-to-date popularity. In this case, the contents in caches have to be regularly updated to prevent information becoming outdated, and at the same time users' requested files must be delivered. These two tasks are challenging in practical heavy-traffic and multiuser scenarios when the network resources are limited. In this paper, we apply non-orthogonal multiple access (NOMA) to facilitate concurrent caching replacement and content delivery in downlink transmission. We formulate a resource allocation problem to investigate how to efficiently push proactive files to the cache at the small base station and deliver the requested files to users. The resource-allocation problem is formulated as a mixed-integer exponential conic optimization problem. To enable a computationally-efficient optimal solution with finite convergence, we develop an iterative algorithm based on polyhedral outer approximation, where a polyhedral relaxation subproblem and a convex subproblem are constructed and iteratively solved to tighten the lower and upper bounds for the optimum, respectively. The numerical results demonstrate significant performance gains of the NOMA-enabled data transmission scheme in power and resource savings compared to the baseline scheme.