A typical Simpsons episode "takes 6-8 months to do one episode" according to Simpsons creator Matt Groeing. Of course, they are creating multiple episodes at a time, but from the starting ideas to finish episode, it takes months.
The average time between the last game of the NFL Conference Championship and the Super Bowl is 2 weeks or less. In 1999, the Conference Championship was January 23rd, and the Super Bowl was 7 days later, on January 30th.
This was the day before the Simpsons episode "Sunday, Cruddy Sunday" aired on January 31st.
The reason they cover their mouths, out-of-universe, is simple. They wanted to air an episode about the then current Super Bowl, without the complications of redrawing the scenes in a week, or multiple times. By hiding the mouths, they could quickly dub the team names in. Alternatively, they could have animated the mouths to fit multiple team names, but that would be much costlier.
According to Wikipedia, who cite the episode writer Mike Scully's Commentary for "Sunday, Cruddy Sunday", in The Simpsons: The Complete Tenth Season DVD set:
Originally, the characters would be saying something else, however, because the staff wanted the episode to be "current", new dialogue was recorded for the scene. Because there was no time to animate the scene from scratch, the staff simply made the characters hold a glass in front of their mouths while saying their lines.