Assuming I'm understanding correctly, seems easy to me.

    .sidebar h4 {
      color: red;
      font-size: 20px;
    }
    .box .box__header {
      color: green;
    }

<div class="sidebar"><div class="box"><h4>Header</h4></div></div> will produce red text.

<div class="sidebar"><div class="box"><h4 class="box__header">Header</h4></div></div> will produce green text.

Both will have the 20px font-size.

It all depends on how you plan out the HTML structure and CSS selectors.

EDIT: I would also say if you are placing HTML like this into the page in several places, then the CSS for it should be self-contained and totally separated fron the rest of the page. I would be reluctant to use default CSS on a container such as that. But I can see it would depend on the nature of the container.