You can't do this no. There is one attribute selector that matches exactly or partial until a - sign, but it wouldn't work here because you have multiple attributes. If the class name you are looking for would always be first, you could do this:
<html>
<head>
<title>Test Page</title>
<style type="text/css">
div[class|=status] { background-color:red; }
</style>
</head>
<body>
<div id='A' class='status-important bar-class'>A</div>
<div id='B' class='bar-class'>B</div>
<div id='C' class='status-low-priority bar-class'>C</div>
</body>
</html>
Note that this is just to point out which CSS attribute selector is the closest, it is not recommended to assume class names will always be in front since javascript could manipulate the attribute.