Unfortunately, rotating a viewport title is not as cut-and-dry as one would expect. When selecting the viewport title, the Rotate command is inaccessible. Fortunately, there is a quick fix available
We need to remember that a viewport title is a Revit system family. In other words, the viewport title is created by the program in the project environment. Therefore, the first step is to find out which component families are embedded in the viewport title. One can verify the latter by selecting the Viewport Title, and editing the Type Properties. In the Title row, we can see that the M_View Title component family is used.
By editing the M_View Title family, one will now be able to rotate the View Labels, and draw a line which will indicate the new viewport title line. Note that this viewport title line will not be editable in the project environment, so think about a realistic view title line length for all views that will be rotated on your sheets. (One obviously needs to save this family as "M_View Title_Rotated", to ensure you are not overriding your other viewport titles)
When loading the modified View Title family into your project, you will notice the new rotated view title line, complete with the original viewport title line still being visible. Look at the Type Selector in your Properties Panel. The Viewport type is set to "Title w Line".
Simply change the Viewport Type to "Title No Line" and your rotated viewport problems will be something of the past.