Issue syncing iPhone 4 with iTunes
It sounds like you have a couple of options to try.

Approach one: First, backup your wife's iPhone 3 in iTunes. Then plug your iPhone 4 into iTunes and on the initial screen for your device select Restore. Select her most recent backup. This should put all her settings as well as apps on your phone. After this you can change settings to suit you best

Approach two: Turn on iCloud on your wife's phone, and plug it in and lock it. After it is backed up. Wipe your phone (Settings > General > Reset > Erase All Content and Settings Warning this will completely delete everything off of your phone!!). Then, when you boot it up, click restore from iCloud and sign in to your Apple ID there.

It is important to note that all her settings (including email, etc.) will be synced in either of these methods. You can obviously change that after the restore, but both methods should provide all the apps and such. I'm not familiar with how it would work having two iPhones with the same Apple ID in the long run so there may be other bumps down the road with this process.

Depending on where songs were bought, you may, with her phone plugged in, have to select File > Transfer Purchases from "her name's" iPhone. It will prompt you to login with your Apple ID, and should transfer the music to the computer. From there, you should be able to just copy the music onto your phone. Otherwise select the Music tab with your iPhone 4 plugged in and select the music you would like to sync to your phone.