"pull it on" refers to the action of putting on an item of clothing, so in this sentence, it is meaning the common/familiar actions we are all used to doing regularly. As Gary said "pull on it" has a different meaning, it would refer to the action of using force to move something. I would also say that, in my opinion, "to pull on a cigarette" is not a common expression but rather the following verbs would be better used: --> "to drag on a cigarette / to take a drag on a cigarette" --> "to puff on a cigarette / to take a puff on a cigarette" --> "to inhale the smoke from a cigarette" (this one is very formal and almost scientific though)

Note that it's 'pull it on' not 'pull on it', so it can't be referring to cigarette. It's referring to a shirt. You pull a shirt over your head. Here, it is using a shirt as a simile for fear.