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.