I know there is such a word but I cannot recall it. This word can be used figuratively when saying that someone gets surprised, angry or react badly to something.
The word bristle is used to talk about animals' hair standing stiffly because they are afraid or angry. And it is also used figuratively to say that someone is angry or annoyed. For example:
The director bristled at the fact that the movie got negative reviews.
The phrase get someone's back up can also be used to say that someone is made angry or anoyed or someone makes someone angry or annoyed. This phrase is, obviously, a reference to cats, that is when they are angry, they arch their backs. For example:
The comment about the boss's weight got his back up.
A person can also "get their hackles up", or you can "make someone's hackles rise". In the literal sense, hackles are the bristly hairs on the back of an animal (particularly a dog), which stand up when the animal is alarmed or frightened. Humans don't have hackles in the literal sense, but one can still talk about a person who is alarmed, annoyed, or frightened "having their hackles up."
Usage examples from the OED:
His voice was tinged with enough condescension to make my hackles rise.
A blinkered assumption... led them to behave in a manner which undoubtedly got the public's hackles up.
I don't think there is a word for it as such, but the most common expression I am aware of is "arching their back".
From the website petassure.com:
Why do cats arch their backs? ... Not only does he arch his back as a form of stretching "sleepy" muscles after a nap, the arched back is also a form of showing that the cat is feeling threatened. In the latter case, the arched back is usually accompanied by his hair standing out all over his body, especially on his tail.
The figurative expression you may be thinking of is "get (one's) back up", which means (in humans) "to become or cause to become angry, hostile, defensive, or irritable".
"an unduly fearful person"
I recall this word getting used as a child, usually chanted like an insult when someone was afraid to do something.
Occasionally in a sentence like:
He won't do it because he is a scaredy-cat.