I've heard this way of talking about dates sometimes, but it is rare, and I'm not sure, if it is correct German. (And I'm a German native speaker.) For my feeling of German it sounds wrong, but I can't name you the rules.
If you are learning German, I suggest, that you should not learn this construction. Better use this:
An den vier Sonntagen vor Weihnachten ...
Remember, that the object after the preposition »an« can be in dative or accusative. If it is a place where something is located (or a time where something is happening), then use dative. If it is a target that will be reached, then use accusative.
In this example something is happening at this four sundays, so you must use dative, which means, that the article has to be »den« (plural dative), not »die« (plural accusative).