SHANGHAI—China is poised to realize an ambitious mission to the far side of the moon, the most immediate of many planned milestones in its effort to challenge America’s half-century-long supremacy in space.

In a first for any country, the Chang’e-4 probe is set to touch down on the “dark side” of the moon on or around Jan. 3, according to state media, and dispatch a rover in a vast crater to explore the moon’s interior. While impressive in itself, the mission is a step toward bolder objectives: China plans to operate a manned...