I've heard a non-native English speaker use the expression "in this place, having a long arm to scratch someone else's back is more important than having a long résumé", conveying the idea of favoritism in a place influencing the promotion of people within. I would like to know if there is an equivalent in English to such expression.
It's not what you know but who you know.
seems to have similar connotations, although is more about knowing the right people, than currying favour with them.
There is the more salacious
Sleep your way to the top
which takes the back scratching a little more literally.
I understood your question as asking for an idiom for nepotism or sycophantic promotion.
One could say, "When it comes to promotions in this company, flattery will get you everywhere."
There's a theory on the origin of "Bob's your uncle" which is based on nepotism (although the saying doesn't mean that anymore).
At first I misinterpreted the quotation, thinking it to mean, "it's more important to help other people than to look after yourself;" that is, employees should be looking out for others – particularly others within the company. In the U.S., many quotes promoting such an unselfish attitude are borrowed from the world of sports, such as:
Everyone is needed, but no one is necessary. (Bruce Coslet, American football coach)
but the most famous I thought of was:
There is no "I" in team.
Getting back to your question, some have parodied that expression, to put it more in line with the underhanded dealings of a workplace such as you describe:
"There is no "I" in team – but there is an "M" and an "E"!
implying that, at the end of the day, you've got to "look out for #1."
It's not a full maxim, and I think it may be an Americanism, but the verb phrase buck for can be used:
You know that and I know that, but somebody's bucking for a promotion. (from the movie Fletch)
All that said, I don't think any of these are as apt as Matt's "it's not what you know, but who you know" suggestion.