Celeborn was of the Sindar, a kinsman of Thingol, so he had never been to Valinor, yet he was entitled to go there like all elves. Galadriel had been born there, being a Noldo, and wanted to go back. But she had lived with Celeborn for three ages of Middle-Earth. It seems wrong that she just dumped him.
To answer your question we must first figure out who Celeborn (Sindarin: "Silver Tree" or "Silver-tall" ... not to be confused with Celeborn the White Tree of Tol Eressëa) was.
Would you like to look behind Door Number One a.k.a. The Silmarillion version? In which Celeborn was a Sindarin elf living in the Maia Melian's realm of Doriath where Galadriel met him.
“Galadriel his sister went not with him to Nargothrond, for in Doriath dwelt Celeborn, kinsman of Thingol, and there was great love between them. Therefore she remained in the Hidden Kingdom, and abode with Melian, and of her learned great lore and wisdom concerning Middle-earth.”
How about Door Number Two a.k.a. The Lord of the Rings Book Two Chapter Seven version? In which Celeborn was a Nandorin elf already living in Lórien where Galadriel met him.
"He has dwelt in the West since the days of the dawn, and I have dwelt with him years uncounted; for ere the fall of Nargothrond or Gondolin I passed over the mountains, and together through ages of the world we have fought the long defeat."
Or perhaps Door Number Three a.k.a. The Lord of the Rings Appendix B version? In which Celeborn was a Sindarin elf of noble blood living in Círdan's realm where Galadriel met him.
"In Lindon south of the Lune dwelt for a time Celeborn, kinsman of Thingol ..."
Options 1 and 3 are a tad bit hinky because Galadriel was also closely related to Thingol, through her mother Eärwin who was the daughter of Thingol's brother Olwë. Celeborn was the son of Galadhon, the son of Thingol's brother Elmo. Which would make Celeborn and Galadriel kissing cousins, so to speak.
Okay, after all that genealogical pondering where are we? Oh yes, your question "Why did Galadriel abandon Celeborn at the end of the Third Age?"
In a word, because she was finished with Middle-earth. Galadriel had accomplished everything possible for an elf in the Third Age of Middle-earth, including single-handedly casting down Sauron's fortress of Dol Guldur.
More than that, Galadriel had at last pierced through the seductive illusion of the idea of rulership itself. She had come to Middle-earth in the ancient days before the rising of the Moon or Sun because "... she yearned to see the wide unguarded lands and to rule there a realm at her own will."
Yet this was the identical trap into which first the Vala Melkor and then the Maia Sauron had fallen into. Their dreams of rulership had also started out nobly, in honor of their conception of the design of Eru Ilúvatar. In her wisdom, and after untold millennia of bitter experience, Galadriel was able to overcome this alluring but ultimately unfulfilling desire. Thus she earned the pardon of the Valar and was allowed to take ship to Valinor.
But Celeborn wasn't ready to go.
For one thing, while Galadriel yearned to return to the home of her birth Celeborn was a Moriquendi elf who had never been to Valinor and was quite content living in the land of his birth. While unlikely, it's possible he may have been old enough to have been among the elves who refused Oromë's summons to the Great Journey.
Less charitably, at the start of the Fourth Age of Middle-earth Celeborn had delusions of grandeur wherein he would split the rulership of the forest of Mirkwood, renamed Eryn Lasgalen (Sindarin: "The Wood of Greenleaves") with King Thranduil.
“But after the passing of Galadriel in a few years Celeborn grew weary of his realm and went to Imladris to dwell with the sons of Elrond. In the Greenwood the Silvan Elves remained untroubled, but in Lórien there lingered sadly only a few of its former people, and there was no longer light or song in Caras Galadhon.”
Despite his vaunted wisdom (Galadriel: "For the Lord of the Galadhrim is accounted the wisest of the Elves of Middle-earth ...") Celeborn remained unable to see the proverbial writing on the wall. That the age of the elves in Middle-earth had come to an end and that they were doomed to "... dwindle to a rustic folk of dell and cave, slowly to forget and to be forgotten."
There is an old saying that may apply here: "We can be knowledgable with another person's knowledge but we cannot be wise with another person's wisdom."
So Galadriel had no choice but to leave Celeborn to his folly until he too drank the dregs of bittersweetness from his alloted cup of life. Tolkien gave us a foreshadowing of this in Celeborn's parting words to Aragorn II Elessar in Fangorn Forest:
"Kinsman, farewell! May your doom be other than mine, and your treasure remain with you to the end!"
Besides, Galadriel was romantically involved with Gandalf the White (see Could there ever have been a romantic angle between Galadriel and Mithrandir(Gandalf)?). I'm betting she spent a lot of time in Gandalf's cabin on that long, sweet cruise to Tol Eressëa.
Celeborn eventually wised up and, as usual, followed Galadriel's lead.
"... at last he sought the Grey Havens, and with him went the last living memory of the Elder Days in Middle-earth."