If I'm correct, from your previous questions, this is your son who will be turning 1 in January? He won't know about, or ever remember, or comprehend this birthday in any form or way. Sure, he may enjoy patting his hands in frosting, or the sound that ripping paper makes for a few minutes. First birthdays are for the mom, the dad, the grandparents and maybe the siblings. You can give your child a mini cupcake on his actual birthday and let his sister give him a new toy. If relatives live nearby and want to come over, just have a simple dinner with a baked ham, or a lasagna, or simple hors d'oeuvres and let them take photos of the baby and take turns holding him and just enjoy him, on any day that's convenient (maybe the middle or end of January!).
I'd like to offer some advice, in the kindest way possible. Many of your questions have been about birthday parties, invitations, RSVPs, dates, etc, and you sound anxious. And now you're stressing about dates for an infant's birthday. Might I suggest that you relax about birthday parties? Don't try to keep up with everyone. The birthday party circuses are becoming more and more complicated and expensive and stressful and out of control. Tell your children that when they turn 10 (double digits!!) you'll throw them a big party with a bounce house and invitations and a clown or whatever, but until then, parties will be simple celebrations. Go to a circus or an IMAX movie or somewhere you wouldn't usually go, with just your family. It will still be special. But the dates aren't set in cement. If your daughter's birthday is in June but Disney on Ice is in town in July, and that's what she loves, take her in July for her birthday and buy her a souvenir. If you relax the standards now while they're young, you'll save yourself a tremendous amount of headaches. This is just my opinion of course. Feel free to ignore it!
Have simple family celebrations. Enjoy your children. When your kids are 1 or 2 or 3, just keep things simple and family-centered. A one year old doesn't need decorations, or invitations, or festivities on the actual birthday date. On our first child's first birthday, my husband made our favorite dessert (his and mine, not our baby's), and we bought one little cupcake for the baby to play with. My husband and I popped a bottle of champagne at the exact moment of his birth and toasted each other. The baby really enjoyed banging on the box that the toy from his grandparents arrived in.