OP, don't you have 13 hours? So let's say with a nap at your hotel and travel/airport time, you'll have 5-8 hours to see the city.

Overall, I agree with what deniah wrote, but some additions or alternatives:

You can't possibly actually see everything, except from the outside. There's a tram line that runs around the Ring. You might consider taking the tram and just getting off for specific sights high on your list, or if something catches your eye. But walking is certainly nice too.

I've never used the CAT, but as I recall it's on the east side of the Ring, near the Hilton. The tram is just a short walk, as is the center. Most of the big sights (IMO) are on the south and west of the ring, as well as in the center. Take a look at a map of Vienna and decide for yourself. I'd say, walk from the CAT/hotel to the St. Stephen's in the center. Lots of good little cafes or sausage stands along the way, and take a look inside the cathedral. Then walk to Michaelerplatz, then down past the National Library, and enjoy the nearby parks. All the museums and the Opera House (worth a visit IMO) are nearby and just off the tram line that will take you back to your hotel.