If my son had apnea of prematurity, how long will I need to have a monitor

Usually not needed. Preterm infants with apnea usually outgrow this condition prior to discharge from the hospital. If they do go home on a monitor it is usually for parental, not medical, indications in which case the duration of monitoring is quite variable. Most infants discharged on home on apnea monitors are taken off by 6 months of age.
No specific length. The use of a monitor continues until there is a significant period without apnea.
Downloads . The routine kid on a monitor will generally stay on it until s/he has two negative downloads (no significant apnea or bradycardia), on no medications, over an 8 week period. The average runs about 4 months of age, but those with odd findings on downloads deserve the extra monitoring.