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An extensive experimental investigation on the structural, static magnetic, and non-equilibrium dynamical properties of polycrystalline Mn$_2$CuGe Heusler alloy using powder X-ray diffraction, DC magnetization, magnetic relaxation, magnetic memory effect, and specific heat measurements is presented. Structural studies reveal that the alloy crystallizes in a mixed hexagonal crystal structure (space groups P3c1 (no. 158) and P6$_3$/mmc (no. 194)) with lattice parameters a = b = 7.18(4) $\mathring{A}$ and c = 13.12(4) $\mathring{A}$ for the majority phase. The DC magnetization analysis reveals a paramagnetic to ferrimagnetic phase transition around T$_C$ $\approx$ 682 K with a compensation of magnetization at $\approx$ 250 K, and a spin-glass transition around T$_P$ $\approx$ 25.6 K. The N\'eel theory of ferrimagnets supports the ferrimagnetic nature of the studied alloy and the estimated T$_C$ ($\approx$ 687 K) from this theory is consistent with that obtained from the DC magnetization data. A detailed study of non-equilibrium spin dynamics via magnetic relaxation and memory effect experiments shows the evolution of the system through a number of intermediate states and striking magnetic memory effect. Furthermore, heat capacity measurements suggest a large electronic contribution to the specific heat capacity suggesting strong spin fluctuations, due to competing magnetic interactions. All the observations render a spin-glass behavior in Mn$_2$CuGe, attributed to the magnetic frustration possibly arising out of the competing ferromagnetic and antiferromagnetic interactions.