My DS was 4 mths at his first Christmas, and to be honest we didn't do anything special. He had a present from us but we didn't do a stocking, having spent so much on baby stuff already that year. They get so much from family and friends anyway.
What he did like was having his regular feeds and sleeps, and lots of attention. In the run up to Christmas, he enjoyed seeing all the bright lights and colourful window displays of the shops, particularly if there were moving figures. You could take DD around garden centres looking at the displays, some are quite extensive.
You could buy a 'Baby's first Christmas decoration for the tree. You could fill a little stocking with practical things she will need, such as feeding spoons, baby toothbrush, new dummy, baby bubble bath, socks, bib, small toy, teething toy. (It would never have occurred to me to do this when DS was small) Have sparkly, colourful decorations.
For presents, look for things that have multi sensory features (colours, noises, textures) and attached mirrors, teethers, flaps to lift and so on. Toys that attach to car seat and pushchair are useful for a long time and also less likely to get lost. It would be nice to wrap a little present from DD to DP, doesn't need to be expensive. I think your DP might come round now there is DD.
What you do on the day is entirely up to you, if it's just the three of you, have a nice cosy relaxed day together. You don't even have to have the traditional meal, have whatever you fancy that doesn't need lots of work.