Complicated iOS upgrade - no wifi, no computer
You can use any computer that has iTunes installed on it. All you need to do is back it up to that computer first. You don't have to sync it with the computer.

Once the iPhone is backed up, you can download and install the update knowing you have a backup of it on the computer.

Once the iOS update is complete, you can go into Preferences -> Devices and then select the backup and delete it from your computer if you don't want it on your computer.