I have been asked a few times by friends how to sync their iPhones to another (or a new) computer without having iTunes delete the apps they already have on the iPhone. Well, the answer is quite simple really.
Connect your iPhone to the new computer
SInce you are connecting your iPhone to the computer for the first time, iTunes will ask you to authorize the computer. You are only allowed to authorize your iPhone to 5 different computers. Here’s what happens when I connect my iPhone to a PC for the first time and launch iTunes:
So all you have to do is go to the iTunes menu, click on Store and then Authorize Computer.
Now (using iTunes on a Mac as an example), right-click on the iPhone on the left pane of iTunes and click on Transfer Purchases.
Your iPhone apps will now be copied over to the iTunes library.
You’re now done!
If you wish, you can then sync your iPhone Contacts, Calendars, Mail Accounts, Bookmarks and Notes to the new computer. Select your device in iTunes and click on the Info tab. Select the appropriate checkboxes you wish to sync (Contacts, Mail, etc.).
If your computer is new, make sure you DO NOT replace the information on your iPhone by unselecting the checkboxes in the Advanced box.
Click on Sync.
I hope this helps!