In this paper, we conduct a theoretical analysis of how to integrate reconfigurable intelligent surfaces (RIS) with cooperative non-orthogonal multiple access (NOMA), considering URLLC. We consider a downlink two-user cooperative NOMA system employing short-packet communications, where the two users are denoted by the central user (CU) and the cell-edge user (CEU), respectively, and an RIS is deployed to enhance signal quality. Specifically, compared to CEU, CU lies nearer from BS and enjoys the higher channel gains. Closed-form expressions for the CU's average block error rate (BLER) are derived. Furthermore, we evaluate the CEU's BLER performance utilizing selective combining (SC) and derive a tight lower bound under maximum ratio combining (MRC). Simulation results are provided to our analyses and demonstrate that the RIS-assisted system significantly outperforms its counterpart without RIS in terms of BLER. Notably, MRC achieves a squared multiple of the diversity gain of the SC, leading to more reliable performance, especially for the CEU. Furthermore, by dividing the RIS into two zones, each dedicated to a specific user, the average BLER can be further reduced, particularly for the CEU.