What is Irish national sport?

The sport you are referring to is called Gaelic football, which is one of Ireland's national sports. There are a few things in it which you would find in Rugby or in soccer, but it is a totally unique game in its own right with its own rules and history. So it would be wrong to say it is rugby and football together. Many ball sports have things in common with other ball sports, but they are unique. That is true of Gaelic Football.