I would literally render litterarum vetustatem as 'the old of the writings,' why is it in the genitive? Because Cicero put it in the genitive, as you can see in the translation below, the translation can be smoothed over for more idiomatic English.
If memoranda have the same force and authority, and are arranged with the same care as accounts, where is the need of making an account-book? of making out careful lists? of keeping a regular order? of making a permanent record of old writings?
With naufragia fortunae, it could be the Genitive of Indefinite Value, but truly, it's just the "shipwreck of fortune" plain and simple no matter what.
Here is the Loeb translation of the whole sentence:
Can I therefore, if I have granted him that bodily pains are counted evils, that shipwreck of fortune is counted evil, be angry with him when he says that not all good men are happy, since the things which he reckons as evil can come upon all good men?
I feel the following statement from Ohio State's Classics Department's page on Latin grammar may be of use to you here:
"I have not listed all the kinds of genitive identified by grammarians. There comes a point when such exercises in categorization become self-serving and even an obstacle to understanding."
In short, it doesn't matter too much, the important thing is that you read the Latin. Cicero's (or any Latin writer's) work, does not lose value if you skim a two word phrase and keep going. It's in the genitive because it's in the genitive, it could be on purpose, it could be a mistake.
I have spent 20 minutes on a clause before to determine if they verb was 2nd Plural Future Perfect Active Indicative, or 2nd Plural Perfect Active Subjunctive, when, in fact, they both translate nearly the same, and it had no practical effect on anything. I understand that drive to know the grammar of something with certainty, but my advice is to keep moving forward.
So if you want to translate it as "old writings" or "fortune of shipwreck," do it! Do what you have to so that you can get to the next sentence.