FAQ / Did Tolkien write a book about the Fourth Age
No, there is no such "book"...
In the 1950s Tolkien began writing such a story, but soon abandoned it again. It seems he took at least two more attempts lateron, but each time lost his interest in it again after a short while.
The extant - rather short - fragments of the story are called "The New Shadow", and the it begins 105 years after the destruction of the ring. They are published in The Peoples of Middle-earth, Vol XII of The History of Middle-earth-series.