Has an NBA game ever ended 87-81?
It has. 87-81 has been a NBA final 37 times in 77,318 games. The first was Chicago Stags beat Indianapolis Jets on January 29, 1949; the latest was New York Knicks beat Indiana Pacers on March 14, 2017.
That makes 87-81 the 643rd most common final score out of 3,189 that have ever occurred in NBA. One point away, 88-81 has happened 42 times and 87-82 has happened 46 times. At one point the score disappeared for 28 years, between 1952 and 1980.