We study the ionization of Rydberg atoms in a single-cycle-pulse electric field based on both classical and quantum calculations. The ionization-probability curve exhibits a "ripple" structure as a function of the pulse duration and the field amplitude. These "ripple" structures are found to be dependent on the angular distribution of the initial state. A large electron-emission asymmetry is observed, and the ionized electron is almost completely emitted to one side of the atom except when the pulse length is roughly one Rydberg period. In both the long-pulse and the short-pulse regimes, larger electron energy can be expected from the ionization of lower-lying Rydberg states, matching the observation in a recent experiment [S. Li and R. R. Jones, Phys. Rev. Lett. 112, 143006 (2014)]. This trend is closely related to the electron emission asymmetry associated with the field-direction change in a single-cycle pulse. The possible implications of the different energy transfer in a single-cycle pulse from that in a multi-cycle pulse are also discussed briefly for the strong field ionization of the ground atomic state.