I know why your Handler does not post delayed.
It is because your stopTime is 0.
You need to set a value for "stopTime", otherwise it will be 0. For example to delay the Runnable for one second:
stopTime = 1000; // milliseconds
Or use your Constructor:
public SongEndCompletionListener(Context activity, String nextView, int time) {
this.nextView = nextView;
actualActivity = activity;
this.stopTime = time; // you forgot this
}