I was also expecting it to be about much older texts, but I guess that was a reading comprehension failure on my part since it did say "books" and not, for example, "stone tablets". As someone who has been (very, very slowly) reading the translations of ancient Sumerian texts, I was kinda hoping this would be about those. But Sumerians didn't write "books".
You got any suggestions for the Sumerian works? I've read The Literature of Ancient Sumer, and made it through, but didn't enjoy it at all and nothing stuck. Attempted From Distant Days (Akkadian literature) but just couldn't do it. Book was poorly put together and it was pretty much the same stuff as the Sumerian book I'd just finished, which wasn't a huge surprise but man that book is bad.
I read all three volumes of Lichtheim's Ancient Egyptian Literature first, though, so that may have spoiled me. It's dry as hell but way better than either of those other volumes. Was good enough that I remember quite a few details and individual works from it years later, unlike those.
I've got about seven volumes of ultra-early literature (the last 1.5 or so of Lichtheim aren't that early), six of which I've read or attempted (haven't made it to the Indian philosophy sourcebook yet, some of which is quite old) and of them the only one I'd recommend to the general reader is Stephen Mitchell's Gilgamesh, which was entirely excellent and so wildly better than any of the other Mesopotamian work from that time that I suspect Mitchell embellished heavily (IIRC reviews indicate he didn't, but still) or the modern version of the work was much improved by the time any of the copies or large fragments we have were written/chiseled. It's crazy good. A couple later Egyptian tales were of similar (though definitely lesser) quality but absolutely nothing else Sumerian or Akkadian's even got a hint of anything like that, from what I've seen.
I'm not sure I'd recommend the Sumerian or Akkadian collections to anyone, really, unless that's their specialty in which case they don't need me to recommend things to them. It's possible I just chose poor collections, though. Certainly the Akkadian one was no good, though I'm not sure there's any better in English.
I've listed the books I have below. I haven't read much of them yet (its slow going). The one that I'm currently in the process of reading is "Myths from Mesopotamia" by Stephanie Dalley. I like it so far, but its not an easy read. It gives you a relatively direct translation of the Sumerian texts, so there are many missing fragments (tablets found in different cities tended to have slightly different versions of the same stories, so you couldn't use them to complete missing bits from tablets from another city). This makes it time consuming to read, but does give you an accurate picture.
The books I have and am slowly trying to get through are:
"Myths from Mesopotamia" by Stephanie Dalley
"History Begins At Sumer" by Samuel Noah Kramer
"Guide To Understanding Sumerian, Assyrian, Babylonian, Cananite And Phoenician Tablets, Slabs, Symbols And Cuneiform Inscriptions" by Maximiliien de Lafayette
"Cuneiform" by Irving Fincel and Jonathan Taylor
"Sumerian Mythology" by Samuel Noah Kramer
"Introduction to Sumerian Grammar" by Daniel Foxvog
I'm not far enough through all of them to be able to properly compare and haven't read any other books, so I'm not sure if I would recommend them or not.
The Cuneiform book was fun (if you're interested in Cuneiform) as cuneiform is easy to write and I was able to write my name phonetically :)