You can use linked text boxes to make all the days (or dates) change throughout the year. In InDesign you can link text boxes from one page to the next.
So depending on which type of calendar you are doing, you either change the day of the week on January 1st, or you change the date to another day;
There is one text box per date and you move the 1st using Enter (force to top of next text box). Here you change where the 1st falls.