The big celebration of Christmas is on Noche Buena, Christmas Eve. Most of the restaurants will be open and busy. Reservations might be required. Be prepared for a celebration quite similar to New Years Eve. Mexican children receive gifts from the three wise kings (reyes magos) on 6 January.

Your closest RC church (Heart of town?) might be Nuestra Senora del Carmen on 15th between 12 north and 12 north bis.
They also hold services in the tiny church in Parque Fundadores near the ferry.
For other options just ask a taxista or check Google for times.