What, if any, celebrations or commemorations were done in 1000 AD?

Nothing much happened, except for some worries about the Apocalypse apparently:

...in Western Europe, during the year 1000, Christian philosophers held many debates on when Jesus was actually born and when the apocalypse would actually occur. This caused confusion between the common people on whether or not the apocalypse would occur at a certain time. Because both literate and illiterate people commonly accepted this idea of the apocalypse, they could only accept what they heard from religious leaders on when the disastrous event would occur.

Although there were debates about the apocalypse itself, few people actually understood the consequences of what would happen if the apocalypse occurred. Unfortunately few documents from around the year 1000 exist to actually interpret what people thought would happen, and because of this many scholars are unaware of what people actually felt.

As the idea of 1000 AD was a strictly Christian one, most of the world either didn't know or didn't care about the 1st millennium for the simple reason that Christianity did not have anything like the global reach and influence that it has today. Even much of northern and eastern Europe had only recently been or not yet been converted to Christianity:

Although most of Spain and Portugal was under Islamic control, Christianity was well established in Western Europe by the year 1000 but around this time it expanded rapidly to the north and east. Sweden's King Olof Skotkonung became a Christian in 994 and Duke Mieszko I had begun to Christianize Poland. Hungary became Christian in 1000. In Russia under the rule of Vladimir the Saint, people were rapidly converting to Byzantine Christianity (Eastern Orthodox). In the West, Iceland and recently settled Greenland also became Christian at this time.

(from http://www.history-magazine.com/1000ad.html)

That leaves a fairly small proportion of the world population who were Christians, and they didn't all agree on when one year ended and the next began:

Bede took it for granted that the year should begin with the birth of Christ himself, on December 25th. But following that logic back through nine months of pregnancy, one arrived at March 25th, feast of the Annunciation, or Lady Day....For a Christian this represented the earliest manifestation of the Divine Presence on earth, and Lady Day was accordingly celebrated as the true beginning of the year. As late as the 1660s, Samuel Pepys reflected this enduring confusion in his Diaries, starting his reckoning of the years on Lady Day (march 25) but also noting the Roman consular date of January 1 as "New Year's Day."

(from 'The Year 1000' by R. Lacey & D. Danziger)

Also, the fact that the year of Christ's birth was incorrectly calculated escaped the notice of most people, though not of Bede. However, as noted by sempaiscuba in his answer, some people linked 1000 AD to Armageddon while most of the population just got on with their daily lives. In short, there were no great buildings or monuments erected.

Most people in Christian medieval Europe who gave the matter any thought seem to have expected that the end of the first millennium would also be accompanied by the end of the world. However, it is also undoubtedly true that most people in Europe at this time were illiterate and probably had no idea that the year 1000AD (or M AD, if you prefer your dates in Latin) was approaching until they were told about it by their local clergy. Those people were kept busy with the daily business of staying alive.

The Bible had predicted how the world would end - with a detailed description of the battles, disasters and plagues which would precede the descent of the new Jerusalem, from the skies - but somehow omitted the exact date.

Over the centuries, theologians had studied the Book of Revelation, and tried hard to calculate a date. By far the most obvious date was the year 1000 (although this wasn't universally accepted).

However, not unlike the last millennium, where most celebrations began a year early, on New Years Day 2000, people seem to been confused about when the first millennium ended. Not everyone realised that the first millennium actually ended at the end of 1000AD, rather than the beginning.

We're told that huge crowds gathered in Rome and Jerusalem on New Year's Eve in 999AD waiting for the end of the world. These pilgrims were generally from among the wealthier levels of society. Many would gather there again a year later. After all that effort and expense, I suspect that some were probably bitterly disappointed when it didn't materialise.

The Pope (Sylvester II), blessed the crowd in Rome, on each occasion and sent them home. The Patriarch of Jerusalem did likewise. The people went home and got on with their lives.

The events surrounding that millennium in Christian Europe were neatly summarised by the chronicler Rudolph Glaber. Glaber was a monk at the abbey of Cluny, and wrote of the build-up to the first millennium a few years later. He described it thus:

"For it was as if the whole world were shaking itself free, shrugging off the burden of the past, and cladding itself everywhere in a white mantle of churches"

Obviously, that wasn't the end of attempts to predict the end of the world. Joachim, the Abbot of Fiore, later calculated that the end would come in the year 1260. In that year, he wrote, the final epoch of human history, "the epoch of the new spiritual order", would begin with a bang. It seems he was mistaken as well.

Source:

Holland, Tom: Millennium: The End of the World and the Forging of Christendom, Hachette , 2011