The story was published in the April 10, 1981 issue, two days before the launch. It was rather short; when it was collected in The Winds of Time and Other Stories, it only filled four pages. I wish I could share the story with you, but the estate still owns the rights, and my attempts to get it reprinted in a current electronic magazine were unsuccessful.
But I can summarize the story for you. Virginia Ratner is the pilot chosen for the last shuttle for the Terrestrial Space Agency. She's been piloting shuttles for 20 years, and the program is coming to an end. She doesn't want to be the last pilot, as she is nostalgic for the program, but she takes the job and pilots the last shuttle to leave planet Earth.
But the story, although bittersweet, is hopeful. For the reason this is the last shuttle is because the human race has colonized the rest of the solar system, and we're giving Earth back to nature:
Earth was returned to its wilderness and its wildlife by a humanity grateful to its mother planet and ready to retire it to the rest it deserved. It would remain forever as a monument to humanity's origin....
Earth was free! Free at last!
I imagine another universe, in which Asimov were still alive, and asked today to write a story about his thoughts as the last shuttle mission came to an end.
Somehow, I can't imagine him feeling as hopeful as he obviously did back when he wrote "The Last Shuttle."