Sync and backup iPhone to new computer?

my Mac computer's hard drive crashed, and was restored from the latest backup date, which was about 6 months ago, so it is essentially a new computer. I want to update the software on my iPhone, and back it up to the computer. However, I'm afraid that by plugging it in to a computer it does not recognize will erase all content on the phone. When I plugged my iPod nano into the computer simply to charge it, even though I ejected it from iTunes, everything that was on the iPod (music, movies, etc.) was erased. I don't want to loose the content on my phone, so is there a simple way to back up an iPhone to a new computer without loosing any content (music, videos, photos, apps, etc.)?