Solid acid catalysts based on graphite-like mesoporous carbon material Sibunit were developed for the one-pot solubilization–hydrolysis–dehydration of cellulose into glucose and 5-hydroxymethylfurfural (5-HMF). The catalysts were produced by treating Sibunit surface with three different procedures to form acidic and sulfo groups on the catalyst surface. The techniques used were: (1) sulfonation by H2SO4 at 80–250 °C, (2) oxidation by wet air or 32 v/v% solution of HNO3, and (3) oxidation-sulfonation what meant additional sulfonating all the oxidized carbons at 200 °C. All the catalysts were characterized by low-temperature N2 adsorption, titration with NaOH, TEM, XPS. Sulfonation of Sibunit was shown to be accompanied by surface oxidation (formation of acidic groups) and the high amount of acidic groups prevented additional sulfonation of the surface. All the Sibunit treatment methods increased the surface acidity in 3–15 times up to 0.14–0.62 mmol g−1 compared to pure carbon (0.042 mmol g−1). The catalysts were tested in the depolymerization of mechanically activated microcrystalline cellulose at 180 °C in pure water. The main products 5-HMF and glucose were produced with the yields in the range of 8–22 wt% and 12–46 wt%, respectively. The maximal yield were achieved over Sibunit sulfonated at 200 °C. An essential difference in the composition of main products obtained with solid acid Sibunit carbon catalysts (glucose, 5-HMF) and soluble in water H2SO4 catalysts (formic and levulinic acids) as well as strong dependence of the reaction kinetics on the morphology of carbon catalysts argue for heterogenious mechanism of cellulose depolymerization over Sibunit.