Even if you are seated at 5:30 (when Ca. Grill opens for dinner) you may have difficulty making that show. Cirque requests that patrons to be seated at least 30 minutes before showtime (you'll miss the preshow if you don't). Should the full meal take two hours (and it very well might), you're out at 7:30, assuming you got seated immediately. An hour is usually enough time to get to DTD via Disney transportation but...
Unless you have a car or take a cab, when you leave from the Contemporary you will need to grab the Downtown Disney bus. First, you might have to wait up to 20 minutes and perhaps more for the bus to arrive. The bus will go to the Polynesian and possibly the Grand Floridian (GF usually shares with another resort, but they do go there sometimes on the Contemp/Poly bus). Then it's a long ride to DTD. You won't get off the bus at the first stop (Marketplace) but the second stop (PI). From the PI stop you will need to walk across all of West Side to get to the Cirque theater.
If it were me, and I didn't have a car, I would plan on eating somewhere at or closer to DTD before Cirque, or budget money for a taxi. It could be a tight squeeze if you don't. That's assuming you go at 5:30. Go any later and you probably won't make it at all.