The card you don't put into your graveyard stays on top of your library.

If a triggered ability triggers whenever day becomes night or night becomes day, it only triggers if it was already either day/night and becomes the other. It does not trigger when the game becomes day or night for the first time.

Day and night are designations that the game itself can have. The game starts as neither. Once the game becomes day (or less commonly, night), the game will be exactly one of them—day or night—going back and forth for the rest of the game.

Before a player untaps their permanents during the untap step, the game checks to see if the day/night designation should change.

If it is day, and the active player of the previous turn cast no spells during their turn, it becomes night.

If it is night, and the active player of the previous turn cast two or more spells during their turn, it becomes day.