In this paper, we mainly consider reliability inference and remaining useful life prediction for the unit Gompertz distribution and the related stress-strength model. The exact confidence interval for the model parameter $\beta$ is derived. The generalized confidence intervals for model parameter $\alpha$ and some commonly used reliability metrics such as the failure rate function, the quantile, and the reliability function are explored. Based on the observed failure data set, the prediction intervals for the remaining useful life and the future failure times are developed. In addition, when the stress and strength variables follow the unit Gompertz distributions with different parameters, the generalized confidence interval for the reliability of the stress-strength model is also proposed. A Monte Carlo simulation study is implemented to evaluate the accuracy of the proposed inferential procedures and compared with the Wald and bootstrap methods. Finally, two examples are provided to illustrate the applicability of the proposed methods.