There is no good way to do this yet (I am working on a suggestion to get this designed/implemented).
But here is a possible way to do it (undocumented and may stop working at any time):
CopyFile[CloudObject["file.css"], CloudObject["file.css"], "MIMEType" -> "text/css"]
Basically this copies the file to itself and uses the "MIMEType" option that CopyFile happens to support.