Is it possible to update my iPhone 3G without a computer? I want new the new update for my iPhone 3G (used as an iPod) but do not want to mess anything up. Can I update it without a computer? I also have no SIM card for the phone.
No, you can't — you need a computer. Over-the-air updates were introduced in iOS 5. As the iPhone 3G can't run iOS 5, you can't do OTA updates, and therefore you need a computer to update it.

Also, 'the new update' (I presume you are referring to iOS 7) isn't compatible with the iPhone 3G, regardless of how you update.