I don't find a listing for it in Cohen's Long Steel Rail, but it's on the Rounder CD, Train 45: Railroad Songs of the Early 1900's. I haven't time to do a by-ear transcription now, but here's what's in the CD booklet:

southwest Missouri and then Arkansas.
No one seems to know anything about the Pullman Porters Quartette; whether they were really Pullman train porters is conjectural. But this 1927 (made in-Chicago) a cappella rendition of PULLMAN PASSENGER TRAIN has such lively turn-of-the-century barbershop harmony that we can excuse any deceptive use of pseudonyms.
Today's train travelers, with their short stub-like tickets need to be reminded that half a century ago or more, a ticket consisted of consecutive portions corresponding to the segments of the actual travel; thus, the longer one's trip, the longer one's ticket. "Jack," in the last stanza, refers to a locomotive.

I did get one line in a search:

"I'm gonna buy me a ticket long as my right arm."

Transcription coming later, unless somebody beats me to it. There's a download available here (click).
I'm familiar with a different barbershop version of the song that went:

"I'm gonna ride that train,
That southbound passenger train…"

I think the Limeliters recorded it. Another search came up wiht another possible title: "All Out for Birmingham."

-Joe-