This paper presents a numerical technique for analyzing one-dimensional transient temperature distributions in a hollow circular cylinder of functionally graded ceramic-metal-based materials (FGMs) in relation to both the temperature-dependence and continuous/gradual variation of the thermal properties of the FGM, and also the related stress distributions are analyzed. The FGM cylinder is assumed to initially have a steady state of temperature gradient, having exposed to high temperature on the inner ceramic surface and to low temperature on the outer metallic surface associated with its in-service environment. The FGM hollow cylinder is then rapidly cooled on the inner ceramic surface by cold air. The transient temperature and stress distributions in the FGM hollow cylinder are analyzed numerically for a model of mullite-molybdenum system. The analytical technique of temperature distributions shown here is quite simple and widely applicable compared with methods previously proposed by other researchers.