Can I buy Australian dollars at the exchange offices at the Bucharest airport?

Yes, Australian dollars are likely to be available at any exchange counter. They will not have small denominations, so their change will be in local currency. Please visit the exchange counter before arriving at the duty free store (which will take your change as partial payment).

Currency exchanges at the airport always, always take you out, no matter if you are in "the poorest country". Exchanges in the city will give you better rates. For the future, it is best to use debit and credit cards when possible when traveling; assuming you don't have a garbage bank, the charges will be much lower than even a reasonable currency exchange counter in the city center.