The Japanese had used complex lunar calender until 19th century. A month had 28 days then. A month also had 5 weeks, and a week had 6 days. The six days included Taian, Butsumetsu, Senpu, Tomobiki, Shakko, and Sensho. They are called Rokuyo.
Nowdays Japanese people use sun calender, though, the superstitions from the Rokuyo remain. Taian, "the day of great peace," is believed a good day, when most ceremonies are held. Butsumetsu on the other hand is considered the most unlucky day of all since it's "the day Budda died." If you believe in the superstition, anything happening on that day could lead to great misfortune.
Relevance to Urusei Yatsura
Like Tomobiki, Butsumetsu is another day in the lunar calendar that has been referred to in Urusei Yatsura.
Ataru himself was said to be born on a day that happened to simultaneously be Butsumetsu, Friday the 13th, and the 4th day of the month. All three very unlucky signs (the number 4 pronounced "shi" is also an unlucky number to Japanese since it's antonymous with "death").
In the show, Tomobiki High School also has a rival school that is called Butsumetsu High. We never get to see the school, but from what we see of the students, the worst punks and gangs seem to come from that school.
Of course Cherry, being the doomsaying old-fashioned monk that he is, is occasionally pointing out when it's Butsumetsu.