I would check, but I do not believe that you will be lucky. Depending on the price you are willing to pay I would contact the concierge of your hotel or check on Ebay. Regarding Christmas Service in the Berliner Dom I checked and was as unlucky as you. Maybe you would prefer the catholic version in the Hedwigskathedrale:
21:30 | Weihnachtsliedersingen
22:00 | Christmette
Josef Rheinberger: Missa in C-Dur
Weihnachtsmusik von Andreas Hammerschmidt, Georg Friedrich Händel,
Hans Leo Hassler, Michael Praetorius, Felix Mendelssohn Bartholdy, David Willocks
Eugène Gigout: Toccata h-moll
Solisten, Kammersymphonie Berlin
Chor der St. Hedwigs-Kathedrale