Inverted-Neil, from his perspective, opened the unlocked gate, entered, held the gate open, then locked the gate, unlocking it for the non-inverted/forward TP and Ives. He then died after being shot by Volkov after unlocking/reverse-locking the gate.
In the scene where Neil got shot, we see the gate latch or bolt moving (shown for around one second only) showing that the gate was unlocked/reverse-locked by inverted-Neil.
Neil approaches the gate and then enters the unlocked gate.
Neil goes to the left side and holds the gate open.
Neil closes and locks the gate.
Neil gets shot by Volkov.
The latch or bolt (encircled in red in this screenshot) moves, signifying that the gate is unlocked (or locked, in inverted POV).
The part where Neil gets shot is shown at around 04:30 in this Youtube video of the scene.
Animated presentations of the scene:
Another answer here suggests that it was Ives that unlocked the gate. This is contradicted by dialogue among the Protagonist, Ives, and Neil.
Ives: How'd you get that lock open?
The Protagonist: Wasn't me.
Neil: I'm the only one who could've got that door open in time, right, Ives?
Ives: Well, I don't know any locksmiths as good as you.
Ives has nothing to do with unlocking the gate. In fact, he was downed while the gate was being unlocked/reverse-locked by Neil and is not near the lock at all.
Some comments in the other answers: ,  - raise the issue of how Neil "instantly" picked the lock or why no lockpicking was shown in the scene.
To use Ives' words — you have to "stop thinking in linear terms."
Related dialogue from the scene where the Protagonist reverse-shoots a bullet:
The Protagonist: Why does it feel so strange?
Barbara (the Tenet scientist): You're not shooting the bullet, you're catching it.
The Protagonist: Whoa.
In this case, inverted-Neil is not unlocking the gate (in linear terms), he's locking it. Locking a gate is much faster than lockpicking a gate. This is why inverted-Neil wasn't shown to be picking the gate's lock.
Related dialogue from the scene where the Protagonist tries to catch an inverted bullet:
Barbara: You have to have dropped it.
The Protagonist: How can it move before I touch it?
Barbara: From your point of view, you caught it. But from the bullet's point of view, you dropped it.
The Protagonist: But cause comes before effect.
Barbara: No, that's just the way we see time.
The Protagonist: Instinct. Got it.
This shows how good Neil is at lockpicking, that even if inverted, he can unlock locks by instinct.