Even a blind squirrel finds a nut once in a while.
A blind squirrel has no practical option but to keep searching for nuts in spite of its natural limitations. The phrase is often used by the skillful to disparage the lucky breaks of others, but it is also used by the skillful to point out that luck played into their success. Just as persistence and luck intersect in real life, the blind squirrel proverb intersects with the motivational mantras:
Persistence overcomes resistance!
or
If at first you don't succeed, try, try again!
The blind squirrel proverb lands closer to the expression of:
sheer luck
The motivational mantras land closer to the expression of:
try your luck
or
ride your luck
The following are not idioms per se, but they turn a good phrase for the intersection of luck and persistence:
“Diligence is the mother of good fortune.” Miguel de Cervantes Saavedra
"Luck comes and goes; you have to seize it." Jean Van Hamme
“I didn't wait for Luck. I tore after it with a truck.” A.A. Bell
“While persistence offers no guarantees, it does give 'luck' a chance
to operate.” Tom Shippey
“Trust your luck, Taran Wanderer. But don't forget to put out your
nets!” Lloyd Alexander
“Luck is distributed at six in the morning” Luis Zamarro Fraile
“A trifle can be enough when luck is on your side.” Margi Preus
“As regards the extraordinary prizes, the element of luck is the
determining factor.” Theodore Roosevelt
And the following quotes turn a good phrase for the intersection of persistence and success:
“Eighty percent of success is showing up.” Woody Allen
“Fortune sides with him who dares.” Virgil
“Success is stumbling from failure to failure with no loss of
enthusiasm.”
“Kites rise highest against the wind...” Winston Churchill
“People who avoid failure also avoid success.” Robert T. Kiyosaki
“It is hard to fail, but it is worse never to have tried...” Theodore Roosevelt
“There is no failure except in no longer trying.” Elbert Hubbard
“The road to success is dotted with many tempting parking spaces.” Will Rogers
“Develop success from failures. Discouragement and failure are two of
the surest stepping stones to success.” - Dale Carnegie
“Don't be afraid of failure. This is the way to succeed.” LeBron James
“Our business in life is not to succeed, but to continue to
fail in good spirits.” Robert Lois Stevenson
The intersection of these ideas seems to be a popular theme, so if you persist in your archery word picture, you may have stumbled blindly onto an idiom for the future.
Shoot enough arrows, and you may hit a blind squirrel.
or
Even a nut might hit the bullseye.
Or some combination of the two ;) I'll give you a one week head start to register a trademark!
One possibly relevant U.S. English expression is "I was just a stepping stone [for him or her]," where "stepping stone" is understood to be a secure place to set your foot down as you make your way along a muddy walkway, say, or across a small stream. The idea is that you help the person along or across as he or she advances, and then the person leaves you behind.
After helping a person like the one you describe, I've sometimes had the impulse to say "See you next time you need me!" But that would be mean-spirited, wouldn't it?
Best Answer
So what you want is a verb phrase that applies to the person receiving the abuse. Something like "got garbage as thanks for gold", except that that's no good because I just made it up.
"No good deed goes unpunished" doesn't quite apply because it's a whole sentence, and it's used when the person feels like that's the way of the world rather than particular ingratitude specific to a person.
"Bites the hand that feeds them" is perfect except that it puts the focus on the ungrateful person, rather than the victim of the ingratitude. Maybe that's OK, but it certainly can't be directly substituted.
I think the other suggestions so far are also not quite right for similar reasons, and unfortunately my best guess is that there is no such common expression. If one were common, somebody would have thought of it. When I have seen these situations in fiction, the victim might describe what they've done for the person and say "And this is the thanks I get?" This also doesn't fit as a replacement for your phrase, but it might be at least relevant.