I admit that I cannot carry a baby of less than 2 months old because it looks so fragile that I am scared it will be hurt. My mother taught me how when my daughter was born but I really couldn't. Anyway, my mother said that the baby is soft at that age of 1 month so you have to take care of the neck. Don't carry the baby upright, not yet.
You are not the only one, most people are afraid to carry a very small baby, even I was scared, in fact after my son was born. But the nurses and my mother showed me the correct way and encouraged me to hold him and carry him without worrying about dropping him.