Ah I understand now.
So the issue is not that you can't preview a page that's unpublished, but that you can't preview a URL mapped content item which is unpublished.
Each news article on the demo site is a content item of the "News" content type, which is displayed in the detail page when you access the URL map pattern for that content item (none of the news articles has it's own page). However the way URL maps work, if a content item is not published, attempting to access the URL map pattern for that content item will return a 404 rather than show you the detail page. That's just how URL mapping works (I could go into the details of why, but bottom line is that the detail page is still published even when the news item isn't, so page preview applies to the detail page but not the individual content item).
So unfortunately you can't preview a URL mapped content item which is unpublished, at least not directly. Although there are a couple ways you might be able to do this, I hesitate to suggest them unless you're pretty experienced with Velocity, since they would both require some significant coding and testing.
The closest you can come to previewing an unpublished content item is just to view a published one and see how the content from the unpublished item will be formatted when it's displayed. Once you've verified that the formatting on the detail page is correct, you just have to make sure that the values in the fields of each unpublished content item are correct, and expect the detail page to format it correctly once it's published.
Final note: If you're using the Push Publishing feature, you can also fully preview URL mapped content items by just publishing them on the authoring server and viewing them on the front-end of the authoring server before pushing the content item to the production server. But this obviously doesn't help if you have a single server setup.