We investigate when and how the relations of galaxy morphology and star-forming activity with clustercentric radius become evident in galaxy clusters. We identify 162 galaxy clusters with total mass at z = 0.625 in the Horizon Run 5 cosmological hydrodynamical simulation and study how the properties of the galaxies with stellar mass M _* > 5 × 10⁹ M _⊙ near the cluster main progenitors have evolved in the past. Galaxies are classified into disk, spheroid, and irregular morphological types according to the asymmetry and Sérsic index of their stellar mass distribution. We also classify galaxies into active and passive ones depending on their specific star formation rate. We find that the morphology–clustercentric radius relation (MRR) emerges at z ≃ 1.8 as the fraction of spheroidal types exceeds 50% in the central region (d ≲ 0.1R ₂₀₀). Galaxies outside the central region remain disk-dominated. Numerous encounters between galaxies in the central region seem to be responsible for the morphology transformation from disks to spheroids. We also find that the star formation activity–clustercentric radius relation emerges at an epoch different from that of MRR. At z ≃ 0.8, passive galaxies start to dominate the intermediate radius region (0.1 ≲ d/R ₂₀₀ ≲ 0.3), and this "quenching region" grows inward and outward thereafter. The region dominated by early-type galaxies (spheroids and passive disks) first appears at the central region at z ≃ 1.8, expands rapidly to larger radii as the population of passive disks grows in the intermediate radii, and clusters are dominated by early types after z ≃ 0.8.