In this paper, we propose and analyze an unmanned aerial vehicle (UAV)-assisted non-orthogonal multiple access (NOMA) short packet communication (SPC) system, where finite-size data packets are transmitted from a UAV to multiple users. The block error rate (BLER), throughput, and energy efficiency (EE) are used as metrics to evaluate the proposed system performance. We adopt the Gaussian-Chebyshev quadrature and the first-order Riemann integral to derive the approximate expressions of BLERs. The accuracy of the derived expressions is verified by presenting that numerical results are perfectly matched with analysis results. We also formulate optimization problems to maximize the throughput with respect to the number of transmitted bits and the altitude of the UAV. Numerical results in this paper provide useful knowledge about the relationships among BLER, throughput, EE, transmitted packet size, the altitude of the UAV, and the Rician factor.