Searching for title: short story or book from 80's
"Birthdays", a novella by Fred Saberhagen, also the answer to this question; first published in Galaxy, March 1976, which is available at the Internet Archive, click here for download options. It was reprinted in the 1980 anthology Galaxy: The Best of My Years (James Baen, ed.) and in the Saberhagen collections Earth Descended (1981) and Saberhagen: My Best (1987).
The young man wakes up every year from stasis for just one day. He's on a space ship heading for a planet to colonize, I think. The first time he wakes, the computer leads him to a nursery full of babies.
Looking back, Bart could never clearly remember any part of his life before the day when the Ship first woke him from a long, artificially induced sleep, and guided him to the nursery to see the babies. That day and the first few that followed it were very confusing to live through.
Each time he wakes the children are a year older. The computer tells him that he is there to help influence the children and help them develop.
"The prime directives under which I operate are very clear. One human parent, adoptive or real, is necessary to the successful maturation of children; images and machines are psychologically inadequate for best results. Therefore, after receiving some elementary preparation for the role, you will serve as adoptive parent for the first generation of colonists."
[. . . .]
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."
Before the group of kids reaches around 20 years old, he wakes to a new set of babies.
Actually, the last of the children die at the age of 68. When Bart is awakened for the 69th time:
"The prime directives under which I operate are very clear," the Ship said in his ear. "At least one human parent is necessary for children to mature to their full potential.
"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."
With gradual comprehension Bart walked on, guided toward the new nursery by the polyphonic squalling from its full cribs.