Same here. Ironically, it was the first song I thought of when I read ‘banjo’; but I dismissed the thought because it isn’t a ‘cowboy song’.
The song was a plot point in the HBO series The Outsider but I’m not sure if the song itself was heard. (Several people in the comments section for the Youtube video looked it up after watching the show.)
Also, is there a name for that really fast string picking that recurs throughout the song?
Is that a tremolo? I dunno nothin’ ‘bout no banjo playin’, but that’s what Google says.
1963? That banjo player probably had a stack of Dick Dale records.
In my defense, I said it was “old-timey” and might have once been played “on a rinkey-tink piano in a saloon” or “maybe out on the prairie.” A tune can call up "wagon train-way out West” images without necessarily mentioning cowboys. I might think of a crusty old prospector out in the desert or a gamblin’ dandy from back East instead. Like Yancy Derringer, f’rinstance.
The second part of the song (does it actually have lyrics?) is for sure New Orleans-jazzy, but it took me a while to remember it. On the other hand, the first part definitely has an “out West” vibe with the banjo music and he clip-clop rhythm of a horse’s gait. It was hearing the first part on TV or YouTube (I don’t remember what clip was from) that planted the earworm and got me thinking about it.
A riff, maybe?
This song has to be understood in the context of the early-'60s folk revival movement, which it both described and participated in. The folkies came in between the beatniks and the hippies as the youth trend of the day. Folk music was the serious, intellectual trend for those who found pop music too shallow and bebop too abstruse.
I did not know that!
In fact, it has lyrics, describing the folk music scene. Sample verse:
In New Orleans, we saw a gal a-walkin’ with no shoes,
An’ from her throat there comes a growl, she sure was singin’ the blues.
She sang for all humanity, this gal with raven hair.
I said, “It’s for the world to hear, c’mon to Washington Square.”
I remember watching that one as a kid. Blast from the past.
Thanks! I never knew it had lyrics.
Good to know, thanks! 
Wow, I never thought my old-timey earworm would lead to a course in mid-20th century American music!
Do I see Ed “Tomahawk” Ames on that record album?
Yes. Yes, you did.