Hi Pablo,
Rush auto-saves regardless of whether a project sync is enabled. A variety of actions trigger the app to save immediately, including importing or deleting media, adding a title, and of course closing a project.
Sharing/exporting is completely distinct from saving the project file. The purpose of exporting is to create a video file that you can play in a video player like QuickTime or post to a video site like YouTube. The project (which, again, gets saved automatically) is where Rush saves your sequence(s) that consist of a series of source clips and other assets that you've trimmed and perhaps applied various effects to like color correction and speed changes. A project is a living document that you can return to and revise any time.
Whereas the file produced by exporting can be played in many other programs and on myriad websites, your projects can be viewed and edited only in Rush (or Premiere Pro, if you decide to take advantage of that program's more extensive, sophisticated, and flexible set of features.)