You can't sync your iPhone to a different computer as far as multimedia files are concerned. Spend 20 bucks on
MEGAPHONE, it's well worth it. It will allow you to put all your iPhone music, videos and photos into your MBA's iTunes, (and from there, you can sync them back directly to your iPhone), otherwise they will be erased by the empty MBA iTuneslibrary.
As for the migration assistant, I'm not sure what to tell you, except try again. As I remember, I didn't use it. I just copied some of my MBP stuff on DVD's and from there, on the MBA.