How to apply wordpress css styles to a jQueryUi dialog?

You can do it in two ways:

  1. Apply the same IDs, classes and general DOM element identification that you have on your website, to the content being shown in the modal dialog. This way, they’ll read from the same css directives you already have in place.

OR

  1. If the styles you want to copy are on your theme’s style.css then find out their existing counterparts on the markup generated by the modal dialog and then add them to the file.

If the css you want to apply is the one WP uses on the dashboard, then you also have to enqueue it to use on the front-end.