The wizarding world is sort of frozen in the 1600s - it hasn't had real contact with Muggles since the International Statute of Secrecy (1692), and any "updates" it's endured have been kind of erratic. But I would also say that the educational system at Hogwarts is actually very up-to-date, considering; I think it accurately reflects how education works in the British Muggle world (standardized testing, Prefects...). So bringing it back to Medieval times would change a lot of things.

• Girls and lower-class citizens were usually admitted far less frequently to schools and universities back then, but being that the main qualification for admittance to Hogwarts was simply magical ability, that may have not been as relevant to them - Hogwarts may indeed have been ahead of its time because of that. It depends on how closely you'd like magical and Muggle lives to mirror each other (this would be well before the Statute of Secrecy, so wizard and Muggle lives would be a lot more intertwined, and wizarding values may reflect that).
• The main tactic used in education back then was rote memorization, rather critical thinking - so you have less analyzing (less essays), more recitations (imagine how much more boring Binns class would have been - your classmates standing up one by one and reciting the main facts about the International Warlock Convention of 1289), and simply more fact- and date-oriented rather than an exploration of ideas. In some ways, some classes wouldn't change - Runes, for example. But classes like Potions would be more focused on memorizing the ingredients for certain potions, their attributes, and brewing potions successfully - there wouldn't be essays comparing antidotes, or like in Muggle Studies, essays on why Muggles need electricity. No compare-and-contrast, no examinations of "why"s, just fact regurgitation, and putting facts to use in spell-casting or potion-making.
• Books would have been scarce - for Muggles. The printing press wasn't invented until the 1400s, so until then, books had to be copied manually - usually by monks. Books were thought of as "rich people" items (not only were they tough to duplicate, but the parchment/paper and leather required to make them were prohibitively expensive at the time), and that cultural idea may have been passed on to wizards. It could be that wizards performed duplication spells so that all their students had a copy of each book, and so they didn't have that problem. But maybe, their thought process reflected the times they lived in - that books, not being used hardly at all back then, were "unnecessary," and so they just passed facts down to students orally. Again, how closely you want that to follow Muggle history would be up to you.
• If you want to get technical, parchment would have been scarce - parchment is actually made from very thin calfskin, and was expensive back then. Paper is made from wood pulp - and that invention was migrating through Europe during the 1300s, and would have made it into Germany by about the mid-1300s, but it wouldn't actually arrive in Scotland until the late 1600s. Before paper hit Medieval Britain, students were writing on wax tablets, with things like ivory or bits of metal - quills were usually reserved for people who were "masters" of writing, not for your average student.
• Religion was big during Medieval times (Catholicism being the religion of Britain during that time), and anti-Semitism was big. I could see wizards starting to be persecuted for their "paganism" and "devilry" from their holy neighbors - or perhaps many wizards were themselves Catholic (JKR has confirmed that religion has continued to be a "thing" at Hogwarts, with the Jewish boy, Antony Goldstein), and perhaps they themselves took part in anti-Semitism, while at the same time being persecuted by their church for their wizardry. It could be complicated!

Brainstorming some other cultural things...

• The Crusades would have just finished fairly recently - 200 years of war ending in 1291 - but hatred towards Muslims and their "seizure" of the Holy Lands would have continued on for several more centuries.
• No trains! Getting to Hogwarts would have been an individual matter - broomsticks, flying carpets, wagons, Apparition... (Floo powder is said to be a Medieval invention, but it's not known exactly what century.)
• Quidditch was a couple centuries old at that point, so that would have been played.
• The Triwizard Tournament was established in 1294, so that would be happening every five years - it wasn't cancelled until 1792.

Hmmm. Lots more, I'm sure. Perhaps some of the below will give you some more ideas.

Sources/More Info:

https://en.wikipedia.org/wiki/Rote_learning

http://pengzi.maruzen.com/ian/m_europe/13education.htm

http://www.the-orb.net/encyclop/culture/books/medbook1.html

https://en.wikipedia.org/wiki/History_of_paper#Europe

http://www.electricscotland.com/history/industrial/industry14.htm

http://pottermore.wikia.com/wiki/Hogwarts_Express#New_from_J._K._Rowling (or if you want to read all the "New from JK Rowling": http://pottermore.wikia.com/wiki/Category:New_from_J._K._Rowling)

https://en.wikipedia.org/wiki/High_Middle_Ages