Can an apartment complex charge a late fee after you paid rent?

Due ON the first or BY the first?  Usually, and I bet you failed to read property.  It is due BY the first.  That means when the first comes, if the money is not there (one minute after midnight on the first), you are late.  Giving you until the 4th means you have 4 grace days, which means that can accept your payment late without evicting you but it does not mean that you don't owe it BY the first and it does not mean that you do not owe a late fee.  You paid late.  Why haven't you got some back up money?  That's really stupid to have absolutely nothing to pay BY the first, until you get your next pay check, on the 3rd, paying late all the time.  Really stupid.  Wake up and change.  I can't imagine the stress you are under.  It would grind at my nerves.

In almost every case, when your payment is due on a specific date, that's the date you have to pay it--or be charged a late fee. Yours is due on the 1st--not the 4th. Unless your lease actually STATES IN WRITING that you are NOT going to be charged a late fee if you pay before the 4th, then you don't have much choice in the matter. Late is late.  BUT, if I were you, I'd take it to the management and show them you paid the rent on the 3rd. And see if they will refund your late fee. They might if you're nice about it. 

If rent is due on the first of the month, most landlords will activate the late fee on the second, unless there's a grace period that's either written into the lease or mandated by state or local laws. ... Grace periods provide tenants extra time to pay rent before the landlord can legally charge a late fee.

If they did not get the money until the 4th, then your rent is late.  The fact that you tried to send it on the 3rd is not relevant.  The rent is not paid until the LL has the money.

If you paid during standard business hours on the 3rd, they can't charge you b/c of their system or slowness.  "Approved" makes no sense in receiving payments.  It's possible it was not received by their bank until the 4th which means you lose out.  

As far as "I got paid on the 3rd", well, that means you are a month behind in your finances.  You need to have enough in your account to pay your bills on the day they are due even if that means money stays in your account for a couple or few weeks.  Cut your expenses and get yourself on schedule.

They can charge you.  By taking things down to the wire you make extra work for them.   It's due ON THE FIRST. Pay it in time to get there on the first. 

If your rent is due on the 1st then it is late if paid after 1st... if they give you a grace period until 3rd that is nice of them to do that and you need to respect the 3rd is the latest date it has to be cleared and in their account by ...so yes they are correct charging you a late payment as it was late

You paid it late. Now pay the fee.

It's due the 1st, that is all that matters, they are nice enough to normally give you to the 3rd, to have cleared and all that jazz.  But you appear to view the 3rd as an OK time to pay a bill that is due on the 1st. I'd say it is all on you.

This is standard with bills including rent.

If the rent is due on the 1st and a grace period until the 3rd, the rent has to be received (not in processing) by the end of day business day on the 3rd. 

You can ask the fee to be waived, but the complex doesn't need to agree to it.