In the first scene, when shown the first wide shot of the square and city hall, the ball is seen being lowered from the top of the pole, while simultaneously clearly being visible already crashed into the roof (including the cracks in the masonry)
Ned patronizes a USPS post office on New Year's Day, when all such offices would be closed.
Fidel Castro was portrayed as a cigar smoking when this episode aired. In fact he quit smoking in 1986, 12 years before this episode first aired.
Ned Flanders, like everyone else in the U.S. in every year that people have filed their Federal income taxes, would NOT have been able to do so on New Year's Day. The earliest that a person could receive the necessary paperwork (for any filing status), review it all, and then file and mail it in (or more likely, e-file it) would be at least two weeks into a new calendar year.
During the rebuilding of Europe Truman is standing next to Clement Attlee and Charles de Gaulle. De Gaulle was Prime Minister of France yet he left office during the rebuilding of Europe phase. He wouldn't become President of France until 1959. Six years after President Truman left office.
Homer states that Mr. Burns hasn't fired him "even after three meltdowns and one China syndrome". But Mr. Burns fired Homer in Simpson Tide (1998) -- the previous episode.