I am using Windows Phone 1320. Is it possible to download an app or game from windows store through my PC and it automatically gets installed on my Lumia 1320?
Surely you can. All you need is a bit of changes in settings. 1. Go to Settings. 2. Select "Find my Phone". 3. Check mark the first option. 4.Now go to windows phone store through your PC. 5.Download anything you want. 6.That app will automatically get downloaded to your phone.
Assuming you mean the Windows Phone Store (which, at least for now, is separate from the Windows Store), the answer is yes, it is possible. Just go to http://www.windowsphone.com/store and log in with the same Microsoft account that you use on your phone. Then, find the app you wish to install and click the big "Install" button (or "Buy"/"Try" if it's a paid app). As long as your phone has internet connectivity, the app should start downloading and installing in a few moments.
If it doesn't work, go to Settings → "find my phone", and make sure that the option "Always use push notifications ..." is enabled.