Back your phone up to iCloud (this is probably the biggest bottleneck if your data doesn't fit into the free storage). If you have a recent backup, you can wipe your phone (Settings -> General -> Reset -> Erase all content and settings). That should clear up plenty of space on your device. Then upgrade and restore your backup once it's completed. This is how I upgraded to iOS 7 and 8 without having iTunes installed. But definitely be _sure_ you have a backup.