“The man who rolls up his sleeves seldom loses his shirt” means that someone who gets to work to tackle a problem (“rolls up his sleeves”) seldom loses everything (“loses his shirt,” an expression dating from at least the 1880s). “The feller who rolls up his sleeves seldom loses his shirt” was in the syndicated newspaper column “Soda Bill Sez” in March 1949.
 
Google Books appears to indicate that “The man who rolls up his sleeves seldom loses his shirt” is cited in the book Unpopular Opinions; Twenty-One Essays (1946), by Dorothy L. Sayers, but no preview of the book is available. Thomas Cowan is often credited for the saying.
 
   
Google Books
Unpopular Opinions;
Twenty-One Essays
By Dorothy L. Sayers
New York, NY: Harcourt, Brace
1946
Pg. ?:
The man who rolls up his sleeves seldom loses his shirt.
 
11 March 1949, Greensboro (NC) Record, pg. 6B, col. 5:
Soda Bill Sez:
The feller who rolls up his sleeves seldom loses his shirt..
 
28 March 1954, Chicago (IL) Daily Tribune, pg. D24 ad:
CONFUCIUS SAY MAN WHO ROLLS UP SLEEVE SELDOM LOSES HIS SHIRT.
 
16 April 1954, Trenton (NJ) Evening Times, “Quotable Quotes,” pg. 14, col. 4:
“The man who rolls up his sleeves seldom loses his shirt.”—KVP Philosopher.
 
Hathi Trust Digital Library
Catalog of Copyright Entries, 1954
By Library of Congress. Copyright Office.
Washington, DC: U.S. Govt. Print. Off.
1955
Pg. 244, col. 3:
PETTES, ROBERT C.
A man who rolls up his sleeves seldom loses his shirt. [The Gay Philosopher] Col. reproduction of painting. Author of reproduction: Brown & Bigelow. 7May54; H3812.
 
Google News Archive
8 December 1959, Nevada (MO) Daily Mail, pg. 2, col. 1:
He who rolls up his sleeves seldom loses his shirt.
 
Google Books
Lifetime Speaker’s Encyclopedia
By Jacob Morton Braude
Englewood Cliffs, NJ: Prentice-Hall
1962
Pg. 209:
He who rolls up his sleeves seldom loses his shirt.
 
Google News Archive
11 August 1966, New Oxford (PA) Item, “Quotes and Unquotes” by The Editor, pg. 4, col. 2:
A man seldom loses his shirt if he keeps his sleeves rolled up.
 
Google Books
A Kick in the Assets:
10 take-charge strategies for building the wealth you want
By Tod Barnhart
New York, NY: Perigee Books
1999, ©1998
Pg. 73:
The man who rolls up his sleeves seldom loses his shirt. — Thomas Cowan
 
Google Books
Wisdom for the Soul:
Five Millennia of Prescriptions for Spiritual Healing
Edited by Larry Chang
Washington, DC: Gnosophia Publishers
2006
Pg. 30:
The man who rolls up his sleeves seldom loses his shirt.
Thomas Cowan