This book is written much better than you would expect an academic book to be, however this is the only point in it's favor.
One of the most glaring issues is the use of absolutely absurdly idiosyncratic terminology. For instance, she says explicitly that fascism is a type of democracy. This is true using here weird definition of democracy (which, as far as I can tell, is the same as her definition of 'nationalism') but it is not true using anything remotely resembling a normal definition of that word. She also asserts that there are only three civilizations in the world (China, India, and the Monotheistic Civilization). She then proceeds to say virtually nothing about India and focuses almost exclusively on the 'monotheistic civilization.' Again, her definition of 'civilization' is very weird and explicitly does not include, for example, the Roman Empire.
Her second problem involves a weird attempt to psychoanalyze nations and their 'national consciousnesses.' Apparently all of Russian history for the last 400 years is essentially the product of Russia's intelligentsia's envy of the dynamism of western nationalism. Allegedly the reason France became anti-capitalist is because it was so jealous of how well English capitalism was working.
The third main issue is the Eurocentrism. England, France, Russia, the United States and Germany are talked about at length. China and Japan share a reasonable amount of space. Africa and the Middle East are almost entirely ignored (there are a few very brief references to Islam). India (despite being one of only three 'civilizations') is barely talked about. I don't think any of Latin America came up at all. It is rather shocking that decolonization is barely even touched on.
Issue number four: Greenfeld attributes too much to nationalism. Nationalism is obviously an important part of modern history, but it is not the only part of modern history. Greenfeld attributes the French, American, and Industrial revolutions to nationalism. All modern ideologies (from liberalism to communism to fascism) are products of nationalism too. Most technological progress is the product of nationalism. The rise of the west is entirely the product of nationalism, as is the later rise of Japan.
#5: She attributes the origins of nationalism to a historical accident in English history. She goes out of her way to say that *everything* in history is the product of historical accident, but she is putting way too much socio-political change down to some lower class people being promoted to elite ranks in England one time. Also Caspar Hirschi has a pretty compelling book arguing that there were nationalist ideas (although no nationalist movements, or really any significant national consciousness) in the late medieval period, which predates the War of the Roses, which is Greenfeld's beginning of nationalism.