This paper proposes a novel mathematical framework for modeling uncertainties in supOU processes, a common model for long-memory phenomena. We address uncertainties as distortions in reversion and Levy measures, evaluating them simultaneously via state-dependent divergence functions on Musielak-Orlicz spaces. The core of our approach involves solving optimization problems to determine the upper- and lower-bounds of cumulants under a prescribed uncertainty set. Notably, we demonstrate that while classical measures like Kullback-Leibler divergence fail in this context, Musielak-Orlicz spaces effectively resolve these issues. Along with providing sufficient conditions for the well-posedness of these optimizations, we demonstrate the framework's practical utility through a water environmental application, modeling streamflow discharge. This work offers both a theoretical advancement and a robust tool for long-memory process analysis.