Questions & Answers about Завтра будет новый день, и я снова буду стараться лучше.
Because быть (to be) has a simple future form: будет = will be (3rd person singular).
Russian often expresses there will be X with будет + noun phrase, and it doesn’t require a separate word like there.
Both are possible, but they feel a bit different:
- Завтра будет новый день is neutral, simple, and “state-like”: tomorrow there will be a new day.
- Завтра наступит новый день is more “event-like” and slightly more poetic: tomorrow a new day will begin/arrive.
Your sentence chooses the most straightforward everyday wording.