The operational cost of post-combustion carbon capture remains the principal factor impeding its uptake, thus real-time optimization has been proposed for economic operation. This requires that uncertainties be estimated, subjecting its solutions to estimation error. Herein, we propose uncertainty estimation and real-time optimization for post-combustion carbon capture plants. Model parameters are estimated using noisy measurements to address model uncertainty; accordingly, we deploy a low-variance scheme to address noise propagation. Our approach is implemented in a pilot-scale carbon capture plant through uncertain flue gas compositions and thermodynamic activities. The proposed method results in operating points closer to the true optima, with up to 25% improvement in economics compared to alternative operational approaches. Moreover, a robust real-time optimization strategy is proposed for cases in which model parameters and economic factors are simultaneously uncertain. The robust update strategy is deployed jointly with the low-variance estimation scheme to quantify the uncertainty in each model parameter, resulting in economic improvements and a notable reduction (80%) in the set point variability over the standard update approach. Through the schemes proposed, the carbon capture plant was able to operate at set points with high capture rates, low energy consumption, and low cost. This suggests that the proposed approach is suitable for the economic optimization of other energy and carbon capture systems.