For the organ's life. The transplanted organ will never change its genetic identity - it will always be made of different dna than you. Your immune system will ultimately recover from the drugs you took, recognize the organ as "foreign" and reject it. Although a very small percentage of people can get away without taking the meds, we cannot predict who. For this reason, you must plan to take them forever.
Usually life long. Most transplant recipients need to be on immunosuppresive medication for the rest of their lives. In a small number, microchimerism may induce tolerance and allow the person to get off medication.