A catalyst regeneration processing apparatus for an internal-combustion engine includes an electronic control unit. The electronic control unit is configured to determine whether a gap is equal to a predetermined degree or less, the gap being a difference between (a) a progress degree of heat deterioration of a NOx catalyst in a predetermined time in a case of assuming that a regeneration process is executed for the predetermined time and (b) a progress degree of heat deterioration of the NOx catalyst in the predetermined time in a case of assuming that the regeneration process is not executed. The electronic control unit is configured to execute the regeneration process in a case of determining that the gap is equal to the predetermined degree or less, even when a sulfur poisoning quantity does not exceed a permissible upper limit quantity.