Actually, the way it's depicted, it would be more like the Discovery is from Ancient Rome (100 BC) and, after The Burn, they've ended up towards the end of the Dark Ages but still in thick of the Middle Ages in general (830 AD).
The times we've recently come out of are unusual. The 20th Century saw more change than any other century. Culturally, technologically, socially. 1901-2000 isn't a good example to go with. Outside of computer tech, culturally and socially things have slowed down already in the 21st Century and I'd say have backslid in a lot of areas. But that's a whole other topic. Back to this one...
There's not a lot of meaningful difference between the 23rd and 24th Centuries. The ships are twice as fast. Holo-technology is better. AI has advanced. That's about it.
The Dominion is 2,000 years old in DS9, compared to the Federation's 200 years old. Theoretically, even cut off, the Dominion should've been able to completely destroy the Federation, Klingons, and Romulans as soon as the Dominion War started. More time doesn't always mean more advanced. Advancement doesn't always proceed at the same rate. And more advanced doesn't always mean more effective. Otherwise, the United States wouldn't have spent so much time in Vietnam.
So I have no problem believing Discovery could've made it in 32nd Century, with the right context. The ship was also refitted early on in the third season. (Thank you Reprogrammable Matter!) And the crew was trained and educated in what they needed to know. Starfleet kept Discovery's interfaces and layouts the same (even though the guts were completely overhauled), to ease the crew's transition. They explain it in Season 3, Episode 6. It's in the dialogue and in one of the early scenes.