You're not unreasonable to feel that way, provided that you don't demand friends and family live their Christmas by your viewpoint.
Personally I realise that me worrying about Syria or anywhere else won't change a damned thing.
I will admit that I suffer from depression. The "daily-or-more-frequently-thoughts-of-suicide-even-when-I-feel-good" sort of depression. One of the first things my CBT therapist asked me was how often I watched the news. A lot was my answer. I could happily stick a 24 hour news channel on all day if I didn't have better things to do. The thing is, news coverage is OVERWHELMINGLY negative. Disasters, horrific crimes, war - the more graphic the images, the more disgusting the gory details, the more viewers/readers/listeners a news outlet will get. They know how to push your buttons so that you push the button for their channel. In simple maths:
More Distressing Images = More Viewers = KERRRRCHINGG!!!!
It's the same reason that crisis appeals a produced to show the most disturbing images of dying children, maltreated animals or whatever else they are seeking your money for. In the industry it is known as disaster porn - seriously, it really is - and as with "normal" pornography, graphic images are shown with little-to-no storyline in order to elicit a certain response from the viewer and therefore obtain their money as opposed to it being spent on a different "content" provider.
A person would have to be utterly heartless to not be moved by distressing images of war, famine, disease etc. BUT please do not let thoughts of these things take over your life. Give money if you want, buy charity calendars or Christmas cards, even travel out to Syria with a charity to help directly with distributing aid.
But remember, no matter how much you personally, or us all as a country, contribute in terms of money and aid - we cannot prevent somebody from suffering. We can try to help as best we can, but we cannot prevent a war or a natural disaster. We can try to help as best we can, but our worrying won't change a damned thing. More than anything, NONE OF IT IS OUR FAULT.
For me personally, this last point was (and still is) a real struggle for me to recognise. I see horrific images and feel like I have let the victims down in some way, as if I am personally responsible. I might not be perfect, but the war in Syria is not my fault. Floods, earthquakes, hurricanes, none of them are personally my fault. I'd actually prefer none of them to happen.
Try to help the citizens of Aleppo by all means, but don't let the worry be at the cost of your own mental health.