In the absence of external magnetic field and Dzyaloshinskii-Moriya interaction, the magnon excitations in a uniaxial antiferromagnet insulator are usually described by two degenerate magnon modes with the same dispersion but opposite polarizations. In this Letter, we show that these two modes are actually coupled intrinsically through an effective spin-orbit coupling due to the dipolar interaction. The magnon spin dynamics and transport thus display various electronlike spin-orbit phenomena, such as the D'yakonov-Perel'-type spin relaxation and an intrinsic (inverse) spin Hall effect. Our work may potentially broaden the applications of antiferromagnetic insulators in spintronic devices.