The noun spud denotes a potato.
The earliest occurrences of spud-barber that I have found are as follows, in chronological order:
1-: From How Noisy Dick Went: Christmas Prayer by Pidgy Magee, by Malcolm M’Dowell, published in The Chicago Daily News (Chicago, Illinois, USA) of Wednesday 22nd December 1915:
Pidgy Magee and I had met during a chance adventure which took me to the heart of Chicago’s west side lodging house district. Other excursions into the land of hobos, bojacks 1, “mission stiffs” 2 and such types of “down and outs” strengthened our acquaintance and often he was my guide through the devious labyrinths of petty crime, misdirected energies and subjacent humanity. When he worked—an infrequent adventure—he was a “pearl diver” or dishwasher in the cheap eating places of the locality, doubling, when adverse circumstances compelled, as a “spud barber” or potato peeler.
1 The noun bojack seems to denote a tramp.
2 Here, the noun mission stiff denotes a tramp who pretends to be religious so as to obtain free food and lodging.
2-: From a letter to the Editor, by a corporal signing himself ‘Wyreemian’, about troops which were on board the Wyreema, published in The Daily Examiner (Grafton, New South Wales, Australia) of Tuesday 18th February 1919—here, spud-barber is a verb apparently meaning to peel potatoes:
Certainly they were a little drilled body, which is only to be expected, when their training of six to twelve months and longer in the Coastal Battalion, [is?] compared with the “Diggers,” six to eight weeks or less, mostly “Spud Barbering” at Liverpool.
3-: From Hospital Definitions, published in Mountain Mists: A Monthly Magazine Published by the Convalescent Soldiers of Bodington,, N.S.W. (Wentworth Falls, New South Wales, Australia) of January 1920:
The Potato Peeler:– Spud Barber.
4-: From A Dinkum Poacher, by E. Franklin Tregaskis, published in The Australasian (Melbourne, Victoria, Australia) of Saturday 22nd October 1921:
“Excuse my inquiry, but what was your occupation when in Queensland?”
“My first collar was off-side cook and spud barber at a tank-sinking ‘caper’ 3 then I went droving, and have followed it ever since.”
3 The noun caper is a borrowing from Dutch kaper, denoting a privateer, also the captain of a privateer.
5-: From Ranzo: A Sea Story, by Gilbert C. Fry, published in The Sydney Mail (Sydney, New South Wales, Australia) of Wednesday 26th April 1922:
“I had no gay and happy apprenticeship days, like you, and I had no fond and foolish father to stump up a big premium; so, instead of wearing brass buttons and a cap ‘all aback for’ard,’ I went to sea first as a ‘spud barber’ in the boats running across the Western Ocean.”
6-: From Gulls and Ghosts: The Ship that Bill Built, by Jack Hamilton, published in The Herald (Melbourne, Victoria, Australia) of Saturday 29th April 1922:
Captain: All hands on deck. Lay along here, men, and disperse the enemy.
A small boy advances.
Capt.: And who may you be?
Boy: Please, sir, I’m Percy Pots, the fourth spud barber.
Capt.: And where’s the crew?
Boy: Please, sir, I’m the crew. All the others left last night, and they wouldn’t take me because I wasn’t in the Onion Society.
7-: From Shipboard Pets, by Jack Hamilton, published in The Herald (Melbourne, Victoria, Australia) of Thursday 1st March 1923:
“The spectre in flesh and blood was the ship’s baker. He had been stirring up the dough after a convivial evening with the spud-barber and others and had been haunted by Jenny’s plaintive wail.”
8-: From the account of a court case, published in Truth (Sydney, New South Wales, Australia) of Sunday 6th June 1926:
By nightfall Eboo was well and truly “inked.”
Eboo made his way back to the vessel, which was lying in Darling Harbor, and after negotiating the gangway safely, encountered the cook in the galley.
Now, whilst wandering about during the day Eboo acquired a very respectable appetite, and he was prompted to demand of the spud-barber food to appease his hunger.
9-: From The Youngest Passenger. Slept Through It All, published in The Evening News (Rockhampton, Queensland, Australia) of Friday 9th July 1926:
One of the most cheerful and mirth-provoking passengers earned the nickname of “Tibby, the spud barber”; he helped the cook to peel the potatoes on the Burwah.
10-: From Three Pennsylvania Boys Await Cash from Home After Adventure Abroad on Standard Oil Tanker, published in the Morning Advocate (Baton Rouge, Louisiana, USA) of Thursday 6th January 1927:
Three youths from Erie, Pennsylvania, and a companion from Chicago who answered the call of the sea last fall for the first time are now at the local Y. M. C. A. waiting for the telegraph boy who will bring them money to return to their homes […].
[…] The youths from Erie were all ordinary seamen, painting and washing woodwork and shining brass. The fourth signed as a second cook, and became the ship’s “spud barber” in the gallery.