Hmm Mixed feelings on this.
I think its worth pointing out, up front, that working class culture is not inferior or wrong. It is different to middle class culture because it is adaptive to different circumstances
I suspect Michael Gove would disagree with me though because he is a bigoted tory
Having got that out of the way, I do agree that an understanding of middle class culture, rules of behaviour etc.. is useful in certain environments. Both to avoid a sense of alienation and to avoid other people's ignorant snobbery. So I agree with him, on that narrow point, at least.
I think it is an indictment of the British Class system that learning a whole new language of interaction and cultural references should be necessary but there you go.
Also worth pointing out that, while anyone might want to "do as the Romans", a working class person finding their way in a middle class environment is adapting herself to the ways of a more dominant group.
So she may feel compelled to understand her new colleagues whereas they are unlikely to attempt to understand her (as a dominant social group- they don't need to, to "get on")
I think that dynamic does lend itself to internalised feelings or inferiority on occasion as well