IPad – How to upgrade the iOS on the iPad 1st generation

There is no way to do that. Unless you already have a copy of the iOS 5.1.1 package for your iPad downloaded, you can only upgrade to the latest supported version for your device.

iTunes 12.7 or later makes this answer into a no - you can’t do it.

Now that iTunes can not manage Apps, you’re not able to side load them over USB.

The only workaround for now is to install a precedent version of iTunes (see this answer) and follow the rest of this answer.

Here's the easiest way to download apps onto an old iPhone/iPad that doesn't support downloading apps through the App Store (if you're getting an error that you need a greater software version to download):

1. On your old iPhone/iPad, go to Settings -> Store -> set Apps to Off.

2. Go onto your computer (it doesn't matter if it's a PC or a Mac) and open the iTunes app. Then go to the iTunes store and download all the apps you want to be on your iPad/iPhone.

3. If the iTunes on the computer and your iPad are both signed into the same Apple ID, and the iPad is connected to the internet, then go on your iPad/iPhone to App Store -> Purchased -> tap on an individual app you want to install.

4. You'll get a message that says "The current version requires iOS 6.1 or later, but you can download the last compatible version," just hit Download.

This method should at least allow you to have most apps on your outdated device. Most of them will work, but sometimes even once you have the app installed on your it still won't run correctly. That's just tough luck. But in my experience that's not too common; apps usually work just fine.