You want the single word, "shirtsleeves." In formal dress, including business dress, a man wears a "suit" indoors. That means, he wears a "jacket" or "suit jacket" over his shirt. When you look at him, you see his jacket and matching pants, and, in the gap where his jacket does not close completely, some of his shirt and his tie. You do not see the sleeves of his shirt because they are covered by the sleeves of the jacket. "In shirtsleeves" means that someone was dressed in his suit, but has removed his jacket, so that you see the sleeves of his shirt. It means that someone has decided to dress less formally or less properly. In my case, when I worked I never wore a suit at all, but I did wear "dress slacks" (pants that look almost like suit pants," a "dress shirt," and a tie. If someone had asked me how I dressed for work, I'd have said "tie and shirtsleeves."

A man in a shirt = A man wearing a shirt that covers his arms and body. A man in shirt sleeves = A man only wearing the sleeve part of the shirt. This means only his arms are covered, not his chest.