I have a jail broken iPhone 4 with several apps. My phone is jail broken and I do not know how to sync it to iTunes. Does anybody know how I can transfer the apps to the new phone?

Are you saying you've never used iTunes before? The purpose of iTunes is to sync music, apps, etc between the computer and the iDevice.

No, I was not saying that I did not know how to use iTunes. The phone is jailbroken and you cannot just connect it to iTunes.

You are mistaken about not being able to connect to iTunes with a jailbroken device. You can connect the phone to iTunes and complete a backup as normal. The only thing you don't want to do is allow iTunes to update the phone, as that would threaten your jailbreak. Backing up, connecting, and syncing are all OK.