Yes, the year 1000 would have been significant both to medieval intellectuals and more ordinary people alike. However, it wasn’t just fear and anxiety, but a sense of hope that often characterized their attitudes towards the coming of the year 1000 and the apocalypse bringing about a triumph of the good and just over evil.
Christian thought then (and to a lesser extent now) is permeated by the concept of eschatology: how the living and (resurrected) dead would be judged at the end of time by God once again. This was one public, generally-accepted response to the issue of theodicy, why the good could suffer and yet the evil flourish in this physical world. Apocalyptism, broadly speaking, held that such a day of judgment drew close, and there is a sense that behaviours and attitudes changed in anticipation for this event: repenting to enter the Kingdom of Heaven, for instance. It was hence a call to action, hope and also belief. We can speak of an apocalyptic ‘time’ (often one that was to arrive unpredictably, very “soon” and “suddenly” as calls to keep one’s faith). Landes notes ‘unusually pious behaviour’ from many kings, dukes, bishops who acquire sainthood thorugh numerous supererogatory, extremely charitable acts, al to avoid the harsh ends that the unrepentant and unworthy would face at the Last Judgment.
However, there was a diversity in how apocalyptic thought manifested, from more catastrophic, devastating violence seeking to cleanse evil from this world, to the peaceful conversion (‘swords into plowshares’) of those committing evil- often secular lords, who if we accept Bisson’s argument, began to exercise increasingly arbitrarily violence in a post-Carolongian world. The apocalypse and the year 1000 was seen as a potential moment for rewarding the saved, through the perfection of earthly society and the arrival of peace and fellowship. What undergirded all these inherently political, not just theological belief, was the idea that the apocalypse would bring about an inversion of hierarchy (with a ‘true Church’ prevailing).
Some historians cite the writings of Gerbert d’Aurillac (later to be Pope Sylvester II) around the millennium, and its lack of apocalyptic rhetoric or references to 1000, as a form of silence indicating a lack of concern for the apocalypse, but there is a way to square how medieval populations were not a purely superficial lot given to sheer panic and terror, and yet grounded their beliefs in theology. Gerbert (and other writers) avoidance of the topics, Fried suggests, is likely to be an artefact of how most bishops and clerics believed that discussing apocalyptic ideas around 1000 AD encouraged dangerous ideas- a rationale also shared by Augustine, whose writings bear a strong antimillenial and antiapocalyptic tone. If so, apocalyptic fears and countermovements can be contained by denying its causes: after all, Christianity was not just an abstracted belief system, but a series of political, hierarchical institutions too susceptible to losing its control at the hands of ‘false prophets’, ‘false Christs’ etc. There are numerous public debates recorded in centres like France between theologians, for instance, as well as some evidence of popular 'assemblies' of individuals seeking to form a covenant with God, although references to these are more fleeting.
What then, when the apocalypse failed to manifest? At the turn of the year 1000, it would be quite clear the lot of your life would still be terrible- you were probably still a farmer with significant obligations to give up your harvest or render significant amounts of labour services to a lord in a time where you would be considered to be inherently inferior by the lot of your birth- not to forget the absence of recourses to justice when you were inevitably abused by lords (on this have plenty of records, so I hope my broad overview suffices).
Yet, this disappointment did not stop beliefs from persisting, although there is some limitation in how those who would probably have a greatest stake (and hope in) freedom from oppression are scarcely represented in the sources, including the pauperes who differed greatly from the formalized, circulated clerical teaching. The strategy of most apocalyptic preachers was hence to postpone the arrival of the apocalypse, such as Adso’s Libellus de Antichristo seeking to assure its readers than yes, the end was yet to come. In other cases, abbots had to warn monks against fixing a set time for the apocalypse’s arrival, but some like Abbo of Fleury believed there was a comptutational error. Calendars that should have ended their (nineteen-year long) cucles with 1006 instead ended with the year 999, 1000, 1001- distortions in expectation of the apocalypse that now had to be corrected. Abbo would publish a correction in 981, arguing that Christ was actually born 21 years earlier than writers like Dionysus believed, and if the actual passion had already fallen in 979, then current expectations were anachronistic and misguided. Others like Glaber, writing after 1000, are anti-Augustinian in interpreting the Revelation in historical, chronological rather than ‘Christian’, more abstract terms: certain signs like the appearance of prodigies near Orleans, a supernova spotted from 1003-6 and later 1009-14, or the ongoing slaughter of Jews were all impending signs of the apocalypse and God’s judgment.
Unfortunately there is a good deal of literature about the astronomical situation (consider Schaefer’s chapter in the book below) I am less familiar with, but the general shift in the historiography is hence to believe that the apocalypse was a strong motivator and source of theological dispute, and yet a more broad-based ‘hope’ in the period. The biggest limitation would be one not just limited to this theme, but in the period your inquiry is in: namely that most written sources stem from royal correspondence or monastic writing, and from more indirect sources of information like the cult of saints rising around figures associated with apocalyptic justice or protection, determining the exact attitudes of medieval people can be hard. However, hope this answer has been somewhat useful.
Sources:
Primarily The Apocalyptic Year 1000: Religious Expectation and Social Change, 950-1050 ed. Landes, Gow and Van Meter, 2003. Chapters by Landes, Fried Christe and Godden are particularly good- there is a great chapter by Umberto Eco (!) too on literature.
The Oxford Handbook of Medieval Christianity, ed. John Arnold (esp the chapter on Fear, Hope, Death, and Salvation)