Install new apps on iPad 1st (first) generation with max iOS 5.1.1
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.