You can use Content Staging OR export and import the webpart. Content Staging will ONLY sync the configuration of the webpart and not the physical ascx and cs files. So easiest way to do this is to select the webpart and in the upper left column click the box with the arrow pointing right (export). Export that webpart and then download it.
Then go to your other website, login, and go to Sites. On the top, click a green button that says Import sites or objects. Click the Upload package button and find the .zip file you just downloaded from the previous export. Next select that file and click Next. On the Objects selection screen scroll down and check the Import files (recommended) and Import code files boxes and click next. This should move the webpart over to your other environment.
Keep in mind, best practice is to create/develop on your dev box and not prod. Secondly, if your sites are not the same version of Kentico it will cause an error and not allow you to import. Thirdly, if your sites are not on the same hotfix, you will see a warning, which you can still override and import.