During the homily this past sunday, The priest chanted the liturgical calendar. Is this a common or universal practice on the first Sunday in January?
It’s optional. It wasn’t done because it was the first Sunday in January. It was done because it was the Epiphany. The priest (or deacon or a cantor) can sing or say it. It proclaims the date of Easter for that year and the dates of certain movable feasts that are dependent on the date of Easter. You can find it at this .link.
This custom is a holdover from before they had iPhone apps that do it automatically.
Diaries and calendars made this proclamation redundant long before Apple or any other company developed electronic devices. It is still a custom in the Church to make this proclamation.