This is an interesting question, because I don't think that the story ever actually says specifically how old he is, so we just have to use clues throughout to guess. When Paul is introduced, it only says that there was "a boy," so no clues there. Later, his governess, or nurse, said that he was "growing beyond her," and that he was too old to be playing on a rocking-horse anyway. He is old enough to speak, reason things out in his head, and think abstractly, as reflected in his conversation with his mother about luck. The story covers over the span of a year, and after Paul gives his mother the money, it mentions that he would be going to his father's private school in the fall.
Based on all of these clues, we can gather that he is at least 5 years old, probably 6 or 7. Children who were being cared for by their nursery governess usually left their care when they went to school, between ages 5-8. And since the story starts with him just barely growing out of that phase, and ends with him preparing to go to school the next fall, that is a good age to put him into. All of this is using inference, or educated guessing, through taking the clues in the text to try to piece together a fact that isn't actually there. I hope that those thoughts helped; good luck!