दुकान आज बन्द है and दुकान बन्द है आज also mean the same although while speaking, different placement leads to different stressing, that may cause subtle difference in tone conveyed
I thought Aaj could be placed anywhere in the sentence, but for this one they want it at the front. Can it be placed anywhere else and still be correct?
You can use anyone from the following, दुकान आज बंद है / आज दुकान बंद है / दुकान बंद है आज.
It should be correct, I've been speaking Hindi my whole life and I've put aaj in specific places in the sentence that would make sense.