It's all in her mind

Couldn't see anyone else mention this, so here it goes:

Kyun-woo is just a figment of the girl's imagination to help her cope with the crushing lost of her love dying. Consider:

She likes writing stories, so she makes up a story of her own
He just happens to meet her on the train on the anniversary of her love dying
The entire ridiculous plot with the runaway solider is just a framing device for her to vocalize her feelings towards loss and how to deal with it (come on, like an entire platoon of soliders is going to be hunting for a deserter in a theme park in the middle of the night, and then fireworks are set off?), as well as being a cool action sequence that she so dearly loves (shown in the scenes where she imagines the movies she would write)
She even diverges off from her main fantasy and imagines Kyun-woo and her in different settings, like her saving him in a machine gun fight or battling him
This is why Kyun-woo always says that he thinks he is helping her get over her pain, healing her pain
At one point she clearly sees a UFO which obviously wouldn't exist
This is why they have to break up, even though she likes him- she has gotten over the pain of loss
The multiple "near misses" when they fail to run into each other or completely improbable ways that they meet- clearly something than an amateur screenwriter would come up with (in her own mind) rather than reflecting actual reality. These are idealized scenes of her imagination, not what actually happened
Kyun-woo always waits for her to call, he never calls her- he only exists when she needs him to
She eventually deals with all of this and writes down the story of herself and Kyun-woo, having it turned into a successful movie (she is the writer not him)
Kyun-woo doesn't have any ambitions or any aim in life because she doesn't need that part of him to be fleshed out- he is just a guy to be here right now

Although it really only makes a lot of sense up to the overtime part. But that can be considered "overtime"- people who need a happy ending like that appended to the story to be able to cope with it. It wasn't in the original story, and it isn't what actually happened.