To be able to get your apartment rental application fee waived you can submit a copy of your credit report and credit score with your rental application.
The rental application fee is used to pay for the cost of running your credit history and rental history so if you can get good references and have a good credit score, credit history etc then most landlords will waive the rental application fee.
Also you might find a house for rent by owner that will let you rent the place without an application fee and some landlords rent houses when you have bad credit as well.
When you apply to rent and apartment or rent a home and fill out a rental application you should hear back from the landlord or property management on the rental application within a few days.
However if the landlord or property manager is really busy then it may in rare cases take up to a week to hear back on the rental application whether or not you've been approved.
If you paid a rental application fee then it likely is not refundable as well and so you would have to pay another rental application fee to another landlord if you get denied the first rental.
You need to have verifiable income, good credit and good references to get approved easily on a rental application.
The landlord wants to make sure they get a good tenant who will pay their rent on time and who will take good care of the home or apartment they are renting.