How in the world would the codes April knows work on the new Enterprise twenty years later?
Presumably, the ideas are that (1) it wasn't just a backdoor to his Enterprise, it was actually in the core OS of all Federation starships and (2) it was never deleted, possibly because someone else who knew about it [Admiral Marcus? Section 31?] managed to prevent it being changed.
Hell, it may not have been April's *own* backdoor, but something he discovered that was already there.
In issue 3 April says:
"I had this control program built into the computer of the previous Enterprise. My Enterprise. It was a failsafe in the event that an enemy took control of the ship. Even if they wiped out my crew, I could retain control of the central computer and all ship systems."
"I gambled that failsafe was duplicated in the new ship, and that my old command code would still work."
Also, the program he uses is called Emergency Protocol 31.