In this study, ultrasonic measurements were performed on a single crystal of cubic PrNi$_2$Cd$_{20}$, down to a temperature of 0.02 K, to investigate the crystalline electric field ground state and search for possible phase transitions at low temperatures. The elastic constant $(C_{11}-C_{12})/2$, which is related to the $Gamma_3$-symmetry quadrupolar response, exhibits the Curie-type softening at temperatures below $sim$30 K, which indicates that the present system has a $Gamma_3$ non-Kramers doublet ground state. A leveling-off of the elastic response appears below $sim$0.1 K toward the lowest temperatures, which implies the presence of level splitting owing to a long-range order in a finite-volume fraction associated with $Gamma_3$-symmetry multipoles. A magnetic field-temperature phase diagram of the present compound is constructed up to 28 T for $H parallel$ [110]. A clear acoustic de Haas-van Alphen signal and a possible magnetic-field-induced phase transition at $H sim$26 T are also detected by high-magnetic-field measurements.
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