I've played many easy fingerstyle versions of "Amazing Grace" on the ukulele, but at the moment this is my favorite. The tab is available for free in the Solo Ukulele ebook by Wlfried Welti . This is a good first fingerstyle solo for beginners.
Tip: On your left hand, your index finger doesn't move much from the first fret, 2nd string (F) so you can keep it anchored there for most of the song. You can see this in the video of Welti playing the song.
In a recent post, I wrote that I hadn't yet found a good version of "Danny Boy" for the uke. I love this song and once created a Pandora station I called "Danny Boy" so I could hear multiple versions of the song. And I'm not even Irish!
I think I bypassed the arrangement by Roger Ruthen, the PDF Minstrel, because I felt it was too easy (labeled grade 1). But I've come back to this ukulele tab of Danny Boy because although it's simple, it's the prettiest version I've found so far. You can download the pdf of the tab for free from his site.
I don't have a video of Roger Ruthen's arrangement, but here is a video of the story of "Danny Boy" to get you in the mood for St. Paddy's Day.
Playing fingerstyle ukulele, with information about the songs and where to find the ukulele tablature so that you can play these songs yourself.