Structural, electronic, and magnetic properties of modified cubic spinel compound LiNi$_{0.5}$Mn$_{1.5}$O$_{4}$ are studied via x-ray diffraction, resistivity, DC and AC magnetization, heat capacity, neutron diffraction, $^7$Li nuclear magnetic resonance, magnetocaloric effect, magnetic relaxation, and magnetic memory effect experiments. We stabilized this compound in a cubic structure with space group $P4_{3}32$. It exhibits semiconducting character with an electronic band gap of $Delta/k_{rm B} simeq 0.4$ eV. The interaction within each Mn$^{4+}$ and Ni$^{2+}$ sub-lattice and between Mn$^{4+}$ and Ni$^{2+}$ sublattices is found to be ferromagnetic (FM) and antiferromagnetic (AFM), respectively which leads to the onset of a ferrimagnetic transition at $T_{rm C} simeq 125$~K. The reduced values of frustration parameter ($f$) and ordered moments reflect magnetic frustration due to competing FM and AFM interactions. From the $^7$Li NMR shift vs susceptibility plot, the average hyperfine coupling between $^7$Li nuclei and Ni$^{2+}$ and Mn$^{4+}$ spins is calculated to be $sim 672.4$~Oe/$mu_{rm B}$. A detailed critical behaviour study is done in the vicinity of $T_{rm C}$ using modified-Arrott plot, Kouvel-Fisher plot, and universal scaling of magnetization isotherms. The estimated critical exponents correspond to the 3D XY universality class. A large magneto-caloric effect is observed with a maximum isothermal change in entropy $Delta S_m simeq - 11.3$~J/Kg-K and a maximum relative cooling power $RCP simeq 604$~J/Kg for 9~T magnetic field change. The imaginary part of the AC susceptibility depicts a strong frequency dependent hump at $T=T_{rm f2}$ well below the blocking temperature $T_{rm b}simeq120$~K. The Arrhenius behaviour of frequency dependent $T_{rm f2}$ and the absence of ZFC memory confirm the existence of superparamagnetism in the ferrimagnetically ordered state.
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