In the documentation, you’ll note that in the General settings section you can select the jquery-ui theme, which applies to the pop-up dialog boxes. So, that’s how you can change the colors for those.
For the main part of the calendar, yes, best way is to simply copy the stylesheet into a subfolder of your theme, and modify as desired. Alternately, you can use the “Customize” feature of WordPress and try to override the CSS styles there (you may have to use something like !important to force override the stylesheet, which is why it’s easiest to just copy the stylesheet and make any changes you want).