I can't speak to exactly what they would have memorized, but can at least speak to education at the time as it pertains to Cicero's works.

In the 18th century, the classical education model was still in full effect. This meant that rhetoric and rote memorization were intensively studied for years of primary school. And the Latin language was a necessity since the Latin authors informed the study of rhetoric, with Cicero being the central figure. And the study of Latin authors persisted into secondary school.

So it's not just that kids would take a Cicero class in high school. You'd be checking out Cicero (and other Latin authors, to be fair) for about 1/3 of your entire education from grades 1-12 (I don't know the history of kindergarten). With rote memorization also being a key element of education, many of these texts could have been almost fully recalled.

There's a great scene at the beginning of the movie The Assassination of Jesse James by the Coward Robert Ford where Jesse James' gang is waiting for the train they're gonna rob. While sitting in the woods, out of boredom one of the guys just starts reciting a Catullus poem. It's beautiful because memorization and recitation were genuinely that big in early education. This guy that would be uneducated and simple-minded by our standards probably read and memorized more classical texts than even classics majors.

Memorizing entire books of the Iliad and Odyssey was common too going into the 20th century. Memory is a thing that can be trained to improve. So students would likely be able to recite or paraphrase thousands of lines of the most important classical texts.