Is there an idiom or an expression to describe a person who cares too much what other people think about him?
frequently used idiom: thin-skinned
Fig. easily upset or hurt; sensitive. (The opposite of thick-skinned. Typically: be ~; become ~; grow ~.) You'll have to handle Mary's mother carefully. She's very thin-skinned.
In psychology, such a person is referred to as a fragile narcissist
See https://thenarcissisticlife.com/fragile-narcissism/ for a further description and explanation.