Hi Igor,
If you do not have access to the server itself to upload a new theme, and your role does not allow you to choose themes then does the theme you are using as your main theme have a customcss box on it's own theme settings page and are you able to access that? I'm afraid if you can't do any of those things (or get someone else who does have access to do them) then I can't think of any other way - but in that case I would suggest you would need to talk to the people who have asked you to do this and the people who set the permissions which are preventing you from doing it 
If you can get at the customcss for the single theme (or get someone with permissions to do so) then you may be able to add some custom css there which targets your course specifically, without affecting the others - it really depends how much you need to be able to change. If you are making small css only changes then that will probably work, if you need major changes including changes to the layout (normally in the php files) then it wont.
Moodle adds a bodyclass identifier for both the course and the category, so you could use those to add some course specific styling e.g. If the url for my course page is http://mymoodlesite/course/view.php?id=1679 I could add some customcss such as
body.course-1679 {
background-color:red;
}
You can target other specific selectors in a similar way, depnding on just how much you need to change
HTH
Richard
