Not sure this applies in your case...
1. Open iTunes and connect the old iPhone.
2. Select the iPhone in the "Devices" list.
3. Click the "Apps" 'tab' in the main screen.
Does the
old iPhone show all its installed apps?
No: I have no idea!
Yes: Select 'File->"Transfer Purchases from wHATEVERyOUnAMEDyOURiPHONE" .' Hopefully, you can disconnect the
old iPhone when the transfer is compete and then plug in your
new one. Since it probably has very few third-party apps, you should be able to use the "Sync Apps" checkbox. You should get a warning dialog stating something like "All the apps on your iPhone will be replaced with those showing in the iTunes list." Which is what I think you want. Click "OK" and call me in the morning...