Real Solitaire supports Game Center leaderboards and achievements. First, make sure you are signed in to Game Center on your device. On current Apple devices, you can do this from the Apple Games app or from Settings > Game Center.
After that, open Real Solitaire and go to Settings. If your version includes an in-game Game Center option, enable it there. Once Game Center is active, supported scores and achievements should sync automatically.
If Game Center still is not working, please review Apple's support resources for signing in, profile setup, and syncing gameplay progress across devices.