As a home coffee roaster I tend to pay close attention to the beans and coffee, but this does not prevent or discourage me from experimentation and enjoyment of coffee in its many forms.
The use of a cinnamon stick strikes me as preferable to ground cinnamon, unless you mix the cinnamon grounds with the coffee grounds and use a fine-mesh press or filter (I almost never brew drip coffee). However, to impart much flavor from a stick you would have to have a prolonged brewing time, which you should not have in a press.
If, on the other hand, you are simply leaving the stick in the mug of coffee (a la apple cider), then you will have a longer infusion period, but the strength of the cinnamon flavor will change over the course of your coffee drinking.
Interestingly, I could not find much on "Mexican Coffee" as described here in an online search. I'd be curious about the traditions and origins of this drink.
During the holidays, I regularly make coffee with the addition of cinnamon, some brown sugar, nutmeg, clove and allspice and serve this 'spiced coffee' with cream. I use a base of Moka Java.