I found a short sound sample at at Yahoo! Music. (Find the album "Prairie Legend" first, and it will bring up the contents; then you can click on "Where the Sleepy Rio's Flowing." For some reason, it doesn't work when you start by searching for the song.) (And by the way, that price of $14.99 for a 4-CD set sounds too good to be true.)

Here are the words I hear:

... old red gate.
Soon I'll wend my way back to her.
Oh, I pray I'm not too late.

In the instrumental section that follows, you can hear what sounds like a wooden whistle or soprano recorder.

By the way, it must be sort of a musical cliché to call the Rio Grande "sleepy" because I found several other songs that use that phrase:

SLEEPY RIO GRANDE
BY THE SLEEPY RIO GRANDE
DOWN ALONG THE SLEEPY RIO GRANDE
DREAMING ALONG THE SLEEPY RIO GRANDE

In addition, two songs recorded by the Carter Family:
MY TEXAS GIRL, includes the line:
"Where the sleepy Rio Grande goes slowly to the sea."
GOING BACK TO TEXAS includes the lines:
Going back to my good old Texas home, home, home,
Down by the sleepy Rio Grande