Don't want to start all over again, do you?
You are playing through a game, and you don't want to lose your progress, right? Of course not. That's why you save it, but sometimes you have to back that save up. You know, in case you get far in Plants vs Zombies or do some awesome things in Angry Birds.
1) Download AppLinks from Cydia.
2) Close Cydia and launch AppLinks.
3) AppLinks has a circular arrow at the bottom left corner of the screen, tap the arrow.
4) Wait a little while as AppLinks creates links to all of your applications downloaded from the App Store.
5) Use your SSH access to access (how to gain SSH Access) your device to /var/mobile/AppLinks from your computer.
6) You will see a list of all the applications installed on your device.
7) Open the folder that corresponds to your device. (For example, Plants vs Zombies.app)
8) When you open the folder, you will see several files and folders, copy all that content to your computer. Do this for each game you want to save your progress on.
9) Now you can restore your device or update and jailbreak.
10) After resyncing your Apps, use AppLinks again as described in step 3 and again SSH to your device: /var/mobile/AppLinks
11) You can now copy the files from your desktop back into the respective game's folder.
Games normally save your progress under "Documents" but there are some games that save this under Library/Preferences under /var/mobile/AppLinks/AppName. If that happens, you will have to copy all the content of the "Preferences" folder except for the com.apple.PeoplePicker.plist then transfer that content back to the iPhone again once the device has been restored.
Sometimes the folder that contains your "Preferences" will only have that com.apple.PeoplePicker.plist file. That means that the game has its files in a different path. All you can do then is read the names of the files and figure out which contains the save information.