As you know, we use "on" for days: "What are you doing on Christmas day?" (Yes, we add "day" here.) Another natural question is, "What are you doing for Christmas?" This means, "How are you celebrating Christmas?" It's not the same as the "on" question above, because "for Christmas" suggests your plans are connected to the Christmas celebrations. As well as using "at" for time on a clock, we also use it for festivals (ie. points in the cycle of a year): "What are you doing at Christmas?" This can include the Christmas season, not just the 25th of December. Now that we're thinking of Christmas as a season and not just a single day, what preposition do you think fits best in your original question? :)

The best way to say this: "What are you doing at Christmas ?" is "What are you doing for Christmas?"

The best preposition to fill in that blank is "for". "We will be closed for Christmas". But "We will be closed on Christmas Day" would also be good. Your questions are both okay, although "on Christmas" is more likely to refer to just the specific day, while "at Christmas" is more "during the Christmas season".

To fill in the blank, you would say, "Our office will be closed on Christmas." Both sentences are technically correct, but they mean slightly different things. "On Christmas" refers to the actual day of Christmas (24 or 25 December). "At Christmas" would be more general, kind of shorthand for "at Christmastime," so any time in late December.