My sister got pregnant at 19, her boyfriend was 26. Both of them were not ready for a child, it was an accident (my sister was on birth control). My sister decided to keep the baby though not mentally prepared for one. We had grown up without a mother and our father was barely home so she had no idea how to parent. It was really difficult for her, constantly crying and complaining but for her boyfriend, it was different. He grew up with both parents in a loving home so he did extremely well with my niece; working and taking care of her majority of the time while my sister slept. After about a year (with therapy and practice), my sister got better and more comfortable as a mother. It truly takes time.
Woo-Young never had a mother growing up... of course, he doesn't know how to be a mother. Not only that, his father and brother spoiled him to make up for her loss. His character has always been spoiled and immature, it wasn't some sudden change by the author. It would be unrealistic for his character to change so quickly after giving birth. It will be a while before he learns how to parent—this is natural for new parents. Kyung-Joo is much more mature/older, so he will do better than his partner. It is a learning process, if you raised a sibling or a child you would know this. Some of you are so judgmental #-.-)