This paper presents the performance analysis for the two-user downlink cooperative non-orthogonal multiple access (C-NOMA) short-packet communication (SPC) over Rayleigh channels. The near user acts as a relay, which operates in full-duplex (FD) mode and employs decode-and-forward protocol, to forward the information to the far user. Two kinds of residual self-interference (RSI) models, which are modeled as being free of fading and following Rayleigh fading, are considered. We first derive the approximate closed-form expressions of average block error rate (BLER) at both users under these two RSI models. Then, simulations are provided to verify the validity of our theoretical analysis. Finally, we evaluate the BLER performance difference between long-packet and short-packet transmissions for FD C-NOMA and that between FD relaying and half-duplex (HD) relaying. The analysis and simulation results show that, for ultra-reliable and low-latency communication, we can use RSI link modeled as being free of fading to simplify the system analysis. Furthermore, compared to long-packet communication, for FD C-NOMA SPC have a narrow range of power allocation to achieve high-reliable transmission, and because BLER outperformance of FD relaying over HD relaying degrades when the RSI signal becomes strong, the optimal duplex mode for FD C-NOMA SPC is determined by the RSI signal and the target BLERs at both users.