He did, plus a less than perfect knowledge of grasp of the power and tech at his disposal. And he seems to have observed a fair range of time periods on Earth.
But the crew specifically said it was 900 years, didn't they? Well not quite. The person (Jaeger) who identified the 900 year thing was a Geophysicist, not a Historian. Kirk simply agreed with him (and he's not a historian either)
JAEGER: Notice the period, Captain. Nine hundred light years from Earth. It's what might be seen through a viewing scope if it were powerful enough.
TRELANE: Ah, yes. I've been looking in on the doings on your lively little Earth.
KIRK: Then you've been looking in on the doings nine hundred years past.
TRELANE: Oh, really? Have I made an error in time? How fallible of me. Oh, I did so want to make you feel at home. I'm quite proud of the detail.
Given the numerous references in other episodes to anything from the 20th Century as "ancient", it's not that far fetched to suppose that a non-historian would lump all "ancient" things together. It's not that different to today really, when many people treat the Middle Ages as a time of "knights and castles" - in fact it was a period of tremendous cultural and technological change.
So, if Jaeger also considered the trappings around him to be Medieval, a rough figure of 900 years is not that far off. If the far more precise Mr Spock had identified the period, then we'd be in trouble!