The search for alternative methods for the production of new materials or fuel from renewable and sustainable biomass feedstocks has gained increasing attention. In this study, Nypa fruticans (nipa palm) fronds from agricultural residues were evaluated to produce pure cellulose by combining prehydrolysis for 1–3 h at 150 °C, sulfur-free soda cooking for 1–1.5 h at 160 °C with 13–25% active alkali (AA), 0.1% soluble anthraquinone (SAQ) catalyst, and three-stage totally chlorine-free (TCF) bleaching, namely oxygen, peroxymonosulfuric acid, and alkaline hydrogen peroxide stages. The optimal conditions were 3 h prehydrolysis and 1.5 h cooking with 20% AA. Soda cooking with SAQ was better than the kraft and soda process without SAQ. The method decreased the kappa number as a residual lignin content index of pulp from 13.4 to 9.9–10.2 and improved the yields by approximately 6%. The TCF bleaching application produced pure cellulose with a brightness of 92.2% ISO, 94.8% α-cellulose, viscosity of 7.9 cP, and 0.2% ash content. These findings show that nipa palm fronds can be used to produce pure cellulose, serving as a dissolving pulp grade for viscose rayon and cellulose derivatives.