Students enrolled at Princeton at age 13 in the colonial era. Were they at a similar level of academic achievement as today's 18-year old enrollees? Were they considered adults?

The best way to think about the Colonial Colleges (more on them here), which included Princeton, is to think of them as a fairly exclusive boarding school for white boys, attended by boys and men of all ages with some day students. It's an imperfect analogy but it works well enough to communicate what "college" was like in those times. Students' days were fairly regimented and similar day to day - the concept of having a major was still more than a century in the making - and the primary function of their education was to teach them to be smart by teaching them the things that smart men knew. Which is to say, their education wasn't practical in the way we think of a college education today. Rather, they followed what's known as a "classical liberal arts education."

Before we get too far into that and comparisons to today's students, it's helpful to get a sense of what their day looked like. From an older question about student diets at Colonial Colleges:

This online exhibit from Digging Veritas out of Harvard gives a fairly detailed breakdown of their day.

Time Activities

6 am: Morning prayers

7 am: Morning bever (a small meal of beer and bread)

8–11 am: Three hour-long lectures

11 am: Dinner

11–2 pm: Recreation and study

2–5 pm: Meetings with tutors (professors) and study

5 pm: Afternoon bever

6–7:30 pm: Evening prayers

8 – 9 pm: Recreation

9 pm: Retire to rooms, lamps out for underclassmen

11 pm: Lamps out for upperclassmen

This schedule, though, had a whole bunch of caveats based on when we're talking about and who we're talking about. Colleges would raise or lower their admission expectations based on a variety of factors, including their financial situation. When times were flush, a college might include the cost of all food in a student's tuition bill [and make their admission criteria more stringent.] At other times, students would have to pay for their meals, which created a class-based structure where students with limited means had simpler meals than those provided to young men from families of means [and they would accept nearly any boy/man of worthy character who could pay tuition.] Likewise, not all students lived on campus. Many, especially older students, lived in the surrounding community and simply went home for meals.

This can also help us think about the life of a 13-year-old at Princeton. That is, the nature of his experiences would be linked to the reasons he was there and the network around him. If he was attending because his father was in a position of power in the local community, he likely went home and had the responsibilities of a child in his family. If, however, he was attending alone, far from home, he was likely making decisions for himself and had more responsibility. All of that said, age at that time didn't hold the same weight as it does now. There are records of 10-year-olds enrolled at Harvard in that era and it wasn't necessarily a remarkable thing like we think of a 10-year-old at Harvard today. Basically, the adults around the child thought the thing that made the most sense at that moment was for the boy to attend Harvard, so they went. (More here on the history of age as it relates to school.)

In terms of what 13-year-olds were studying at the time, a classical education was heavy on Greek, Latin, some math, some sciences, logic, and rhetoric. While there are lots of benefits to learning Greek and Latin from a learning sciences perspective, they weren't learning Greek and Latin in order to communicate in the language. Rather, it was because smart men knew Greek and Latin. And this, in a nutshell, is the largest difference between a 13-year-old at Princeton in the 1700s and a college First Year today; learning things because that's what smart men knew versus learning things that are applicable to a particular future.

Today's first-year college student is at a specific college because they want to learn specific things that college offers. In effect, they pick a college based on their future plans (or what they can afford, of which college accepted them, etc. But generally speaking.) During their high school experience, they were likely encouraged to pursue topics of interest, to read books that they thought were interesting, and generally encouraged to think about college as a part of finding their place in the world. Their academic achievement can best be defined as "diverse." They studied English, Math, Science, History, Music, Art, physical education, a foreign language, etc. (AKA the modern liberal arts curriculum. More here on the shift from the classical curriculum to the modern one) The math they studied in HS was different than the math a 13-year-old at Princeton studied, not necessarily less or more complex... just different.

Which means, practically speaking, if you put a 13-year-old enrolled at Princeton in 1780 up against a 17-year-old enrolled at Princeton today and the questions were about the Bible and Socrates, the 13-year-old would reign supreme. Ask questions about history, literature, or statistics, the modern kid is [likely] going to come out ahead.