Exact, perfect, perfectly correct, just as it should be, on the button, on the dot.
In the olden days of radio broadcasting, program directors would use silent hand signals to communicate with the performers. To signal that the show was running right on schedule, they would place a finger alongside their nose. It is generally agreed that this is the source of the phrase 'on the nose'.
There are two common folk etymologies claiming that this phrase comes from horse racing. One etymology claims that it comes from the gamblers "betting on the horse's nose", the winning horse being the first to have its nose cross the finish line. Another claims that it's a variation on 'nose-to-nose', referring to two horses crossing the finish line at exactly the same time. Both of these are probably completely false.