Intensive research has been devoted to investigating the reliability of standby systems in the past several decades. The existing models, however, mostly fail to address bounded operation time of the system units (e.g., resulting from limited production resource in practice). In this work, we contribute by modeling and optimizing the mission success probability (MSP) of a 1-out-of-n standby system with a required mission time and heterogenous units, each characterized by its time-to-failure distribution, initial age, and maximum possible operation time. We suggest a numerical MSP evaluation algorithm and solve the optimal unit activation sequence (UAS) problem with the objective to maximize the MSP of the considered standby system. Impacts of mission time and unit operation time limit on the MSP and optimized UAS solutions are examined using a case study of a water supply system for cooling a chemical reactor. Based on the case study, we further demonstrate the joint optimal product resource distribution and UAS problem and the optimal loading problem as well as their solutions of maximizing MSP.