It looks as if Apple has fixed this issue. If you go to the app store on your iOS 5.1.1 device (I used my iPod touch) and attempt to download Spotify, it will now give you the option to install an older version of Spotify that is compatible with your version of iOS. I just did it and now can use my premium account on my iPod touch!
Good luck to everyone else! Hope it works for you too.
P.S. I found this solution from another message link regarding Spotify and older versions.
Hi Guys. I've been reading through the forum for a fix on this. I've just purchased a used Ipad 1 on IOS 5.1.1. One of the things I was looking forward to was spotify. Trying to dowloand it on the app store. A notice appears syaing that i need IOS 6.0 or later for it to work. This is really disappointing of spotify and apple to do this. Why isn't there a version for the ipad 1 ?
What you have to do is log into your iTunes account on your computer and get the spotify app there first, without connecting your iPad. Once you have "purchased" it on your computer and installed it into iTunes (it is okay that it is the latest version), go to the app store on your iPad and get the spotify app. It will offer you a compatible version for iOS 5.1.1. Just don't connect your iPad to your computer during the process and make sure you are signed into the same apple ID as your iTunes account on your computer. You should be all set!