In this answer, I address some moderately pedantic issues that make the answer to the question "No", but I do not address the substantive question of whether that quote is more than 90 years old.
There are several versions of the quotation floating around. So, the first step is to try to find the earliest mention.
Here, I am leveraging off the work of Jim L who addressed this same question on a competing Q&A site. (He concluded that the quote was invalid, but that site has differing standards of evidence than here.)
He found two cites that I have been unable to beat:
A tablet (Assyrian) 2800 B. C. says :
" Our earth is degenerate in these latter days ; there are signs that the world is speedily coming to an end; bribery and corruption are common ; children no longer obey their parents ; every man wants to write a book, and the end of the world is evidently approaching."
Tablet preserved in Constantinople
From here, two points stick out.
The first is "write a book"? This is written on tablets! What is the Assyrian word for book? I'm going to give the benefit of the doubt here to Dawson — perhaps the translation was very literal, and the word "book" had a different meaning.
The second is that it is said to be an Assyrian tablet in 2800 BC.
There is some debate about whether Asyrria existed in 2800 B.C. (for example, a poorly cited Wikipedia entry suggests it was formed when the Akkadian Empire fell circa 2080 BC, while also suggesting it was a part of the earlier Akkadian Empire.)
Whether it existed as a geographical place, there is another question of whether it existed as a language. Jim L., (above), claims, without substantial evidence the Assyrian language hadn't been developed by then.
Andrew George explains that the some earlier Akkadian works, were ascribed to Assyrian:
Because the first substantial discoveries of written Akkadian where made in the ruins of Assyrian cities, Akkadian was known to its first decipherers as Assyrian.
Reference: George , Andrew (2007) "Babylonian and Assyrian: a history of Akkadian". In: Postgate, J. N., (ed.), Languages of Iraq, Ancient and Modern. London: British School of Archaeology in Iraq, pp. 31-71.
[Note: I had some difficulty displaying that link in my browser, but found it worked by downloading it and opening it directly]
In his time-chart (Table 2), George shows Old Assyrian didn't develop until 2000 BC.
By the same chart, it seems suggestive that even Akkadian writings didn't exist in 2800 BC. Certainly, some forms of writing, such as Sumerian did, but I haven't found any references that claim to have samples of Akkadian writing prior to 2500 BC.
- I have not shown whether or not this is a quote from an ancient work.
- I've shown that the quote, and its provenance has survived largely intact since the 1920s at least.
- In particular, it has been traced far further back than Sir Isaac Asimov's book (as suggested by others here).
- However, I have shown it was not both Assyrian and from 2800 BC. It may have in Akkadian, a related language, from 2800 BC, but that is earlier than any references I found so I find it unlikely. It might have been Sumerian.
- IMHO, given the dubious provenance of the source, a more likely scenario is that it is either a true quote, oddly translated, from a much later date, or invented in the early 20th Century.