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The rear sight should have two sizes.  You ought to be able to flip it over and back to choose the size you want to sight with.

The larger diameter opening (Ghost Ring) is for low light and/or closer targets.  You should focus your vision on the target and place the fuzzy front sight which just needs to be within the rear aperture visually on target.  No need to center the front post exactly inside the aperture.  It's more for speed than precision.

Small hole (Peep) is for daylight and precision shots, for both close and long range targets.  You should focus on the front sight, placing it in the center of the rear aperture.

Both rear sight types are normally individually zeroed, meaning there will be a variance in one if you zero the other.  The difference between the two is usually 2.5 minutes of elevation, or 2.5 inches at 100 feet.

Apologies if you already knew this info, but you seemed surprised at the size of the rear aperture.

https://thenewrifleman.com/the-ultimate-guide-to-the-ar15-iron-sights/

Good explanation of your A2 sight vs. the A4 and how everything works:

From what I've read, 1:9 is best for 55-62 gr range, with decent/good in the 45-77 gr range. 

Sorry, can't share personal experience with 1:9 as all of my ARs are 1:7.  I can see if any of my shooting buddies have 1:9, or even 1:8 twist barrels.  For my 1:7 barrels, I have had good results in the 55 to 77 gr range. 

Hit me up if you want some 62 gr and heavier stuff to try out.   

To the question you asked -- if the barrel says 5.56, then you can use either .223 or 5.56.  The weight (grain) of the bullet is more a function of the twist rate.  That ought to be marked somewhere, normally near the muzzle.

This photo shows marking "5.56 NATO 1/7". 




Here's a good chart on twist rate vs. bullet weight.  Works for both .223 & 5.56 ammo.
If you check the chart, you'll see 1/9 is optimal for 55gr and 62gr ammo, as Drk100 said.

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That looks like an older Armalite "National Match" rear sight. They had a funky 1/3 MOA per click
55s should be stabilized with a 1x9 longer ones like the 77s are happier with faster twists.

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Thanks for the feedback....and the chart.
I am familiar with the AR a little. I just gave up on the 223 long ago.
The flip sight is  DEEP welled...not like a standard sight. The larger hole is like the small hole on standard sights and is marked 0-2.
The smaller hole looks like a twin.
First time here...didn't think to look that could enter pics.
Wonder if this is what might be a Match sight.
Barrel stamping is under the guard only.
Anyone know what the XXX stands for? Only see that on premium bolt action barrels....especial Douglas...McGowan air gauged tubes.
And why a 16" HBAR configuration....as most HBARs I seen are 20" tubes.

Anyway, thanks for the replies. Appreciate them much. More of a curiosity as I have dies, brass, heads. But really not into the small caliber. But who knows...might actually be fun...hahhh.
Aloha to all...stay safe.

The pictures help.  I meant to post this video link, but my daughter distracted me talking about tax returns!






The 8/3 dial is the bullet-drop compensation setting.  You dial it to one click above the #3 setting while zeroing, then back to #3 for use.  The BD compensation setting is only used when sighting through the small aperture. Setting 3=300m, 8=800m.  In between settings also correspond to n x 100m.

Zero is usually performed for 300 meters using the smaller aperture.  Once zeroed, the larger aperture will also be zeroed at 200m.  So, you can select a 200m or 300m zero by simply flipping the aperture back and forth.

For zeroing at 300m, set your target at 25m.  That covers the arc of your bullet as it rises through the 25m distance, then drops back down to 300m.  Once zeroed at 25m, you're set for 300m as well.