Yes there was a belief among some Christians that the year 1000 would signal the coming of the Anti-Christ, but it was not a part of the church doctrine. Especially in Anglo-Saxon England it gained some traction.
From The Anglo-Saxon World by Higham & Ryan
"...the growing sense of eschatological expectation in the run-up to the year 1000. Though such expectations were far from universal in western Europe, they formed a significant intellectual current, influential on som Anglo-Saxon thinkers."
Yes. Maybe.
Regarding the year 1000AD:
"In this year a terrible comet appeared, which by its look terrified many, who feared that the last day was at hand; inasmuch as several years before it had been predicted by some, deluded by a false calculation, that the visible world would end in the year of Christ 1000.” -- Annales Hirsaugienses via The Year 1000 And The Antecedents of the Crusades.
But, according to George Burr, that account in Annales wasn't written until the fifteenth or sixteenth century, so who knows. It's likely that many people living in the year 1000AD didn't know what year it actually was and apocalypse literature was so popular back then that in any given year, someone was generally anticipating the end of the world.
"So what if it's the year 1174? Are you saying that our Lord and Savior can't bring about the Kingdom of Heaven just because it's 1174?"
Though it isn't what you asked, exactly, I'll add that Millennial panics in general are really common, though the starting point for those thousand-year periods will vary. The first one that I'm aware of happened in 200AD:
For the first appearance of our Lord in the flesh took place...in the year 5500...and 6,000 years must needs be accomplished, in order that the Sabbath may come, the rest, the holy day “on which God rested from all His works...for “a day with the Lord is as a thousand years.” Since, then, in six days God made all things, it follows that 6,000 years must be fulfilled. And they are not yet fulfilled, as John says: “five are fallen; one is,” that is, the sixth; “the other is not yet come.” -- Hippolytus of Rome
Hippolytus TL;DR: 200AD was supposed to be the Earth's six-thousandth birthday. As God created everything in six days and a thousand years was but a day to the lord, then the world will end in 200AD.
I don't know if anyone took this seriously, but Hippolytus wrote it down, so it must have meant something to someone.
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