This sounds like "Birthdays" by Fred Saberhagen (Galaxy, March 1976).
The protagonist, Bart, is shown 24 cribs by the computer, observes the machines taking care of them, and allowed to pick one of the babies up. He then goes back to sleep for a year:
The machines labored ceaselessly, patting, changing, feeding, washing, wiping up. Twice they dispensed cups of soup-like stuff for Bart to drink. There were no clocks to watch but he was certain that he had been in the nursery for hours. At last, one of the machines took him lightly by the arm and pointed back down the corridor whence he had come.
When he had closed himself into his little plastic-walled bedroom the Ship's voice said: "You will be given a substantial breakfast when you wake again. That will be one standard year from now."
The original children are sterile, and unable to have children. And they find out they were not born of the best genetic material:
"There's further evidence." She paused. "I said all the human seeds and eggs were coded as to type and potential? There's some indication in the available records that all of us now alive—except you, we don’t know where you came from—were conceived from materials not considered of the highest quality. Not that we have any grave genetic defects, of course, no seriously defective material would have been placed aboard. But—not the best. This suggests to me that all the best material was somehow destroyed, and also that there may not be much material left."
After 68 years the last of this generation dies, and he is shown a new generation that he will help raise for landing in 20 years. The computer implies that the entire previous generation has been part of Bart's training.
"We will arrive in less than twenty standard years within a system of planets probably suitable for colonization. From now on you will be awakened increasingly often. You will serve the first generation of colonists as parent. Like them, you have first-rate genetic potential, and perhaps you will remain in some position of leadership when they mature. Today begins your apprenticeship in this role; your elementary preparation for it, a course in the basics of human psychology, was completed yesterday."
You can read the story at the Internet Archive.