This seems to be the original recording. You can hear it at RedHotJazz.com:


YOU'RE BOUND TO LOOK LIKE A MONKEY WHEN YOU GET OLD
(Clarence Williams, Alex Hill, 1930)
(As recorded by Clarence Williams' Novelty Band, Okeh 8798, May-22-1930.)

1. I can tell by your hair
You got monkey in you somewhere—

CHORUS: And you're bound to look like a monkey when you get old*.
When you get old*, when you get old*,
You're bound to look like a monkey when you get old*.

2. I can tell by your feet
You don't get enough to eat—

3. I can tell by your hand
Your pa was a monkey man—

4. I can tell by your jaw
Your pa was your pa—

[*Sometimes he sings "grow old" instead of "get old".]

The Online 78-rpm Discography Project also lists:

YOU'RE BOUND TO LOOK LIKE A MONKEY
Roy Harvey & Jesse Johnson, 1931
Birmingham Serenaders, 1935
Milton Brown & His Brownies, 1935
Jimmie Gordon V-vop Band, 1937
Bob Crosby's Bobcats, 1940

YOU GONNA LOOK LIKE A MONKEY WHEN YOU GET OLD
Robert Hill, 1936.

BOUND TO LOOK LIKE MONKEY
Jimmie Revard & Oklahoma Playboys, 1937