There were few options for America to take in the crisis in Syria. Obama's policy was adequate. Although it wasn't optimal, there weren't really a lot of other options. Romney might have started another middle eastern war, which no one wanted. It is not likely that Romney could have come to a better solution than Obama did.
The Republican Party has a history of going to war. Most recently, George W. Bush put us in two wars, The War on Terror, and the War in Iraq. The War on Terror was certainly justified. The war in Iraq, not so much. These wars put our country in even more financial strain than we were, especially after 2008. I believe Mitt Romney may have been talked in to going to war with Syria, and that would have be devastating to are fragile economy. I don't think this would be the right option. So no, I don't think he would have handled it better.