Learn English – What to call someone who, apart from very inappropriate acts, doesn’t care about what others think about him

I can't think of a noun, but you can say "that person is apathetic".

Apathetic at Oxford Dictionaries

adjective showing or feeling no interest, enthusiasm, or concern

I don't have a single word, but an informal American expression would be

someone who can dish it out but can’t take it - someone easily criticizes other people but does not like it when other people criticize him or her

To dish out originates with the sense of serving food to people, but has taken on an idiomatic meaning of meting out severe punishment or criticism. According to Etymonline, this sense of dish dates from 1934. Taking criticism, punishment, or abuse is from the sense of take as receiving or accepting, from which the idiom taking heat also derives.

While this quality is a characteristic of narcissists, in lay usage narcissism refers mainly to self-absorption or vanity, rather than being simultaneously hypercritical and thin-skinned.