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As a Mac hardware lover, I bought an old 64 GB iPad 1 with iOS 5.1.1 on it. Data wiped.
Nice hardware, but ended up stock with no new application not being iOS 6, iOS 7 or more in requirement.
Thought I would have to jaillbreak it to have new apps.
Is there any other way?
When you choose to backup on iCloud you are creating a backup in the cloud, not the MacBook.
You have the choice to Sync to a computer or to iCloud. If you sync to a computer you can also sync over WiFi, which could be the better option for you by the sound of your question.
If you open iTunes, make sure that it is authorised to your account.
Plug your device in.
It will then say. (substitue iPod for iPhone, iPad etc)
Click "Transfer Purchases"
Your device will now transfer all the apps and data to this new computer before it syncs the new data.
If you have iOS 5 or 6 installed on your iPad, you can use the iCloud Backup functionality to transfer all your settings and apps, without the need for another computer. When you set up your new iPad, it will ask whether you want to set it up as a new iPad, or restore from an iTunes or iCloud backup.
Restoring from an iCloud backup will restore nearly everything, with the exception of any saved Wi-Fi or other passwords, which you'll have to re-enter. Alternately, doing a backup and restore from iTunes on a Mac or PC will restore passwords if you encrypt the backup.
Apple's support site has more details in this article: iOS: Transferring information from your current iPhone, iPad, or iPod touch to a new device.
Best Answer
To install new apps really requiring iOS6+, I don't know.
But after a lot of search I found:
You install iTunes if you haven't already.
You pair your hardware with an Apple account, again if you haven't done already.
You find apps with older version of it existing and working on iOS 5.1.1.
You associate it (get it, download it) to your account on iTunes on Mac or PC.
You find the same app on the older hardware (iPad 1 iOS 5.1.1).
Instead of download, you will see Install. Select Install.
A popup will appear offering downloading older version of the app. Click Yes.
Enjoy your new old 5.1.1 compatible app.
Would love app update for older iOS by developers but don't expect it.