How do I download iOS 4.3.1 to my old 3GS
Connect the iPhone to your computer with the USB cable and then update from iTunes.
If you have an iPhone 3G, iOS 4.2.1 is the latest version. If you actually have a 3GS as you say, the latest update, iOS 6, will be applied.
No other way could I just take phone to AT&T and let AT&T do their thing
Ask them. I don't know what they will or will not do.
I don't have a home computer
Then you can't do the upgrade.
Or ask a friend
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