Just assuming for a moment that this case is really for an iPhone 5 and that the measurements are accurate: 2.8" is half an inch wider than the iPhone 4. Let's be generous and say that the case adds 1/4", which means the new iPhone would be 1/4" wider than the current model. The current iPhone screen is 1.96" wide, so assuming the same bezel width the screen could be ~12.5% wider.
Keeping the same resolution would reduce the pixel density to 289ppi. While it's a possibility that Apple would keep the same resolution with a bigger screen, I think it's unlikely. It would make the UI appear larger, it would annoy some people with good eyes, and it would be bad for marketing. I think it's more likely that Apple would try to keep the pixel density the same, and let apps which cannot deal with an increased resolution/screen size run letterboxed.
Conveniently, 12.5% more than 640 pixels is 720. If this hypothetical iPhone 5 kept the 2:3 aspect ratio, this would mean the screen is 1080 pixels and 3.3" tall, for a 3.97" diagonal. However, if existing applications ran letterboxed, there would be no need to keep the same aspect ratio! What if the new phone was 16:9 and 1280x720? It would obviously be of advantage for video playback, but it also makes better use of the front area.
Such a screen would be 3.92" tall, with a 4.5" diagonal (at the same 326 ppi as the iPhone 4). It's not hard to imagine that, with a screen almost a full inch taller than the current one, they could get rid of the physical home button and use an on-screen button instead. This would also explain the "home button with gesture area" rumors. Of course this raises the question of whether apps could use the full screen. It could be that only the video player app can do it. Or that apps can move the home button to a convenient location. Alternatively, a multitouch gesture could bring back the home screen.
Finally, if this case is really genuine and Apple produces such a large phone, I wouldn't be surprised to see an iPhone 4S as a simple speed upgrade for people who like the current form factor. But it's a big if in any case.