The first example that springs to mind is Holden Caulfield from Catcher in the Rye. Finishing high school represents that awkward, uncertain and downright terrifying period of transition from childhood to adulthood. At that age it can feel like the future is laid out already like a trap. Go to college, get a job, get married, have kids, then it's clean sailing 'til death. Many millennials would prefer to stay indoors playing video games, or go travelling, or do a degree in some unemployable field - all of which could be perceived as lack of motivation by their parents.
Holden's basic character arc is avoidance and eventually acceptance. If your character doesn't know what they want to do next they will either flail about trying to find out, or stick with some old routine and try not to think about it. Procrastination can lead a character up all sorts of avenues, as can a sudden moment of flight when the pressure gets to much. A distraction in the form of a call to adventure would be more then welcome, in that scenario. Or perhaps they pursue an adventure all by themselves just to spice their life up a little, before it's too late. It depends on the story.
As for the breakthrough, I think TheTermiteSociety's comment is key: your character needs to find something that's meaningful to them. Maybe it's the people they meet on the adventure and the relationships they form. Maybe they witnessed some sort of injustice, and found meaning in the fight to stop it. Maybe they simply had that moment of realisation that they already were an adult, and that it wasn't such a bad thing after all. It depends on the character and on the story you're telling. But once they've found that meaning, it can be brought back full circle. Your character could go off and fight a dragon, then come back home and see that college application form still sitting on their desk. It had seemed like a monster before, but not so much anymore.
