Glucose is a widely used chemical, food and feedstock. 5-Hydroxymethylfurfural (5-HMF) is one of the platform molecules, which could be applied in chemical and fuel industries. This work presents the possibility of glucose and 5-HMF production from inedible cellulose, which is the main component of renewable plant (wood) biomass via one-pot hydrolysis–dehydration over solid acid catalysts based on Sibunit carbon material. The catalysts prepared via oxidation with wet air mixture, HNO3, sulfonation by fuming H2SO4 at 80–200 °C or combination of the techniques demonstrated high activity under hydrothermal semi-flow conditions. Dependences of both glucose and 5-HMF formation on catalyst acidity and the nature and amount of surface acid groups are revealed. The total surface acidity but not the chemical nature of the groups is responsible for effective cellulose hydrolysis to glucose. On the other hand, sulfo groups demonstrate noticeable activity in 5-HMF formation from glucose. Glucose and 5-HMF can be derived from the polysaccharide with yields up to 75 and 10%, respectively.