Each day in Legends: Arceus lasts 24 minutes, and each real-time minute is one in-game hour.
In-game time is saved; if you save and reboot your game, it will start at the same time. The save block for in-game time is 203A7F34 and stores a double value equal to the number of minutes. The function that converts this to the hour is at sub_71024AB9F0 in version 1.1.1.
There are 4 different times of day, which I will call Dawn, Day, Dusk, and Night. I do not believe that there are official names, but you may see other names such as Dawn = Morning, Day = Midday, Dusk = Evening on various fansites.
Times will be written as 24-hour time. You can subtract 12 to get the 12-hour time if you prefer.
I recorded a video of the full day cycle, which also shows when Night starts and ends [video]. The hour is displayed on the top right. You may notice the hour actually ticks to 24 very briefly, and the game immediately corrects it to 0 [[video](https://youtu.be/bOxToygZIl8