Look, Apple designed iPhone 5 to be used without a computer, and the fact is millions of users, around the world, use it that way with zero issues...they update, restore, etc.
Yes, you can do this without a computer...are there risks? Of course there are. Make sure you have a fast Internet connection...make sure there aren't any firewalls or other software to block stuff...if unsure, go to an Apple store. They will do this for you.
You can re-download all iTunes purchased content, except ringtones, for free. So, you can restore all of your apps, etc.
Good luck.