The staircase is a fundamental component of buildings that requires proper cleaning and maintenance. This paper illustrates the design and navigation framework of the reconfigurable staircase-service robot called sTetro-C. This novel robotic platform has integrated a robust intra-reconfigurable locomotion system with a stable lifting mechanism and a wet cleaning module to enable smooth navigation on both flat and uneven terrain of the building floor and stairs. Moreover, the staircase climbing framework by sTetro-C utilizes a time-of-flight sensor, a LiDAR sensor, and an RGBD RealSense camera for map building and localization. The control system enables the robot to identify, navigate, and move efficiently on the staircase. The mechanical and electronics systems are also presented with detailed stability, force and kinematic analysis of the lifting and wheels systems, respectively. Simulation results in Gazebo, along with experimental trials in a real environment on the building’s staircase, have demonstrated improved stability of the lifting system and enhanced locomotion of the sTetro-C.