B_Louck,
Would you tell us the exact version number of iOS on your 1st gen iPad? See Settings > General > About > Version. Is it 5.0, 5.1, or 5.1.1?
I have a 1st gen iPad running iOS 5.1.1. I was able to install Adobe Reader version 11.0.1 a few minutes ago.
Have you ever installed Adobe Reader for iOS on your iPad using the same Apple ID (that is shown in Settings > Store)? If this is the first time installing Adobe Reader for iOS using the Apple ID, you need to use iTunes to download and associate the app with the Apple ID first.
Please see Reply #6 in the following post.
Re: How do I get a version of Adobe Reader for an IPod touch with IOS 6.1.6
If you have installed Adobe Reader for iOS before, the app should appear in App Store's Purchased section. You can tap the cloud icon to download and install it.
Please try downloading again if you do not see the 2-button dialog for the first time.