"The staff has the book."
Translation:Tá an leabhar ag an bhfoireann.
"Tá an leabhar acu" would be "They have the book," which isn't a literal enough translation for Duolingo. If you meant "Tá an leabhar acu an bhfoireann", that wouldn't be grammatically correct. "acu" means "at them" and is a complete prepositional phrase, so the "an bhfoireann" isn't serving any grammatical function.
"the staff" is the subject of the English verb "have". Irish doesn't have a verb that means "have", and in the structure that Irish uses to convey the same information as the English verb, "the book" is the subject of the verb bí and "the staff" is the object of the preposition ag.
"X has Y" - Tá Y ag X
No, you should not get in the habit of thinking of this as "the book is at the staff", because that's not what it means. As an English speaker, you don't have any confusion about the meaning of "have" in "I have to go now". With just a little bit of practice, Tá leabhar agam will naturally be understood as "I have a book" unless you train yourself to translate it word by word, in which case you will permanently hobble your ability to understand or speak Irish.
If you've spent six months training yourself to force English word order on Irish sentences, then you may indeed feel hobbled. I know people who have studied Irish for over a decade who still mechanically translate Tá na buachaillí ag rith as "Is the boys at running". On a good day, they can even get as far as "are the boys at running". They have never had a conversation in Irish beyond phrases that they learned by rote like Conas atá tú? - Tá mé go maith, because they insisted on trying to parse Irish by using English word order. But the word order is as intrinsic as the vocabulary, and if you are capable of translating tá as "am", "are" or "is", then you are capable of translating the word order, unless you insist that is too difficult for you - that becomes a self-fulfilling prophecy.
Well, if I may offer some help, based on how I handle it. I started trying to translate word for word, and it fell over more and more, taking me by surprise. So I settled for just working through Duolingo and letting it soak in - that after all is how we learned our mother tongue as children. I find it is working, to a degree. Phrases pop into my mind and I find they are right; I use words and then think about why, and again, they are usually the right ones. Duolingo is absolutley brilliant in allowing us to work using our own methods, we can batter away at it as much as we want; we can ask questions as necessary; and take note of the tips in each lesson when given. But the bottom line is keep at it, every day, little by little and it will come. Maybe think of it like riding a bike. You don't think carefully about which way to lean, or how to balance, you feel your way, and it becomes second nature. Likewise in speaking our mother tongue, not many, if any of us, think about the cases and genders, or whether a word is an adverb, preposition etc. We just use the words through acquired habit and familiarity. So I would suggest you just keep bashing away - an hour a day I do - and don't let it get you down. That seems to be working for me. I hope it does for you.