How do I restore my iPhone 4 without backup?
So, I don't think this is necessarily as simple as the current answers make it out to be.    

Technically no, if you don't have a recent backup, you can't restore your phone to it's most recent state.

However...  a backup (in its simplest form) is just: 
- a list of apps you have installed, and where you have them on your screen
- your email / iOS settings
- any data stored in the apps
- photos (unless you have iCloud photo enabled, then photos are not part of the standard backup)

So, given the above...   music is not part of a backup.   Any music you've purchased through iTunes is already part of your account.   You just need to download it again (it's already paid for).   Any apps you've downloaded can be downloaded again as well...   they are already purchased.   If you aren't using iCloud photos, you might lose some recent pictures taken...  but that's going to be about it.    

Almost everything done on an iPhone is easily recoverable without a "backup".   The backup just makes things easier.