Open Veins of Latin America by Eduardo Galeano is not set in Montevideo, but it's by the city's most famous writer and will give you the political and historical context for everything you see in Uruguay and beyond. Galeano was born in Montevideo in 1940 and spent much of his life writing in the Café Brasilero on Calle Ituzaingó.
The Memory of Fire trilogy (also Galeano) — Génesis, Faces and Masks, Century of the Wind — is a mosaic history of the Americas told in hundreds of short vignettes. Read a chapter a day during your stay; you'll understand more about Montevideo's relationship with its history.
Alive by Piers Paul Read is the account of Uruguayan Air Force Flight 571, which crashed into the Andes in 1972. The story is Uruguayan to its core — the survivors were mostly students from a Montevideo rugby club — and the Museum Andes 1972 will make more sense if you've read the book first.
The Book of Embraces by Eduardo Galeano — a collection of short prose pieces, more intimate than his epic histories. Some of the most beautiful pieces are set in Montevideo: the cafes, the streets, the Carnival.