How do Apostolic Christians interpret arguably sexist verses in Sirach?

Have you checked other translations? I've noticed a couple striking differences in the NRSV-CE and the DRA. E.g

A son ill taught is the confusion of the father: and a foolish daughter shall be to his loss.

The verses are also numbered differently, so that is tripping me up quite a bit as I go back and forth.

Note the modifier that is absent in the NRSVCE, which I expect would change your view of the verse.

Leaving the translation differences aside, I'll focus on your question and post commentaries, using Catena and for the Haydock quotations https://johnblood.gitlab.io/haydock/id330.html

1. I can't find any commentaries on 42:13. Also, the DRA has "and from a woman the iniquity of a man." Another odd discrepancy. St. Thomas, however, does cite this verse in warning men to refrain from spending too much time around women, in order to safeguard his chastity.

   A great obstacle to continence arises from extrinsic circumstances, such as constant intercourse with women. We read in Ecclesiasticus, “Many have perished by the beauty of a woman, and hereby lust is enkindled as a fire . . . for her conversation burneth as fire” (9:9). And, in the same chapter, the following safeguard is proposed against these dangers: “Look not upon a woman that hath a mind for many, lest thou fall into her snares. Use not much the company of her that is a dancer, and hearken not to her lest thou perish by the force of her charms.” Again, in another chapter, “Behold not everybody’s beauty; and tarry not, among women, for from garments cometh a moth, and from a woman the iniquity of a man” (42:12). St. Jerome, in his book Against Vigilantius, writes that a monk, knowing his own frailty, and how fragile is the vessel which he carries, will fear to slip or stumble, lest he fail and be broken.

   On 42:14, see Geremia's answer

2. For Sirach 25:24 I found St John Chrysostom:

   "How is it," one could say, "that Scripture calls a helper she who was a hindrance? In fact it says, "Let us make a helper like him." " And I would ask you: how can she be a helper who deprived the man of such security and drove him out of that wonderful existence in paradise, casting him into the tumult of the present life? A schemer does this, not a helper! "Woman," it says, "was the beginning of sin, and because of her we all die." And blessed Paul says, "Adam was not deceived. It was the woman who, being deceived, transgressed." How, then, can she be a helper who put the man at the mercy of death? How could she be a helper who brought it about that the children of God, which is to say all of the inhabitants of the earth, are submerged in death together with the beasts, the birds and all the other animals? Would not the woman have caused the ruin of righteous Job, if he had not been truly a man? Was it not the woman who brought about Sampson"s ruin? Was it not a woman who did her best that the whole Hebrew people take up the worship of Baal of Peor and was slaughtered at the hands of her brothers? And who more than anyone else consigned Ahab to the devil, and before him Solomon, despite his wisdom and fame? Even now, do not women often lead their husbands to offend God? Did not the wise man say, perhaps for this reason, "Any kind of evil is small, compared with the evil of a woman"? "How is it then," you ask me, "that God said, "Let us make a helper like him"? God does not lie." Nor do I say so"never! She was made for that purpose and reason, but like her companion, she did not want to remain in the dignity that was hers. The man was created by God in his image and likeness. Indeed, God said, "Let us make the man in our image and likeness," just as he said, "Let us make him a helper." Once created however, the man immediately lost both of these prerogatives. He knew how to keep neither the image nor the likeness (and how could he have, if he gave himself over to absurd desire, was prey to deception and was unable to overcome pleasure?). To his disgrace, the image was taken from him for all time to come.

3. Haydock has on 22:3

   Loss. "I wish," said Augustus, "I had never married, or that I had died without children." (Suetonius)

4. And on 36:26 he has

   Good, is not in Greek, but the context shews that it is necessary. (Haydock) --- By concord small possessions increase, as by discord the greatest are lost. (Worthington)

5. On 26:15:

   Hedge. Or "stake," palum, (Haydock) on which tents were fixed, Genesis xxxviii. 14. (Calmet) --- Fail. Incontinence will at last ruin her health. (Haydock)

How do Apostolic Christians interpret arguably sexist verses in Sirach?

By this, I mean anyone in the ... Eastern Orthodox ... Communion.

Eastern Orthodox, born and raised.

from a woman comes woman’s wickedness.

To most males, women are physically beautiful.
But physical beauty needn't necessarily go hand in hand with spiritual beauty.

One must be aware of the distinction,
rather than naively assume the two to go hand in hand.

Better is the wickedness of a man
than a woman who does good;

There are two main types of enemies:

• those that openly oppose or attack you,
  physically or otherwise;
• those that deceitfully employ an apparent goodness or sweetness,
  to veil or cover their true intentions.

One must beware of both kinds of dangers.

The pagan Greek philosopher Aesop has a similar teaching in one of his many fables,
wherein a little mouse is frightened of a rooster's apparently savage behavior,
yet drawn to the mild manners displayed by a cat.

From a woman sin had its beginning

To one's body, women are physically attractive.
To one's soul, sin is morally repellent.

Attraction and repellence are antonymic concepts.

because of her we all die

Mothers give life (Genesis 3:20).

Yet, the (physical) beauty of women can also tempt many males
to engage in (spiritually) ugly things,
whose end is death, be it either moral, or physical, or both.

If she [that is, your wife] does not go as you direct,
separate her from yourself.

Allowed in the old covenant; forbidden in the new.

Nevertheless, the solid life advice still stands,
that the two spouses must be attuned to one another morally and intellectually as well,
beyond mere physicality,
lest their marriage be short lived.

It is a disgrace to be the father of an undisciplined son,
and the birth of a daughter is a loss.

The former, spiritually;
the latter, financially.

Even today, after decades of widespread employment of women,
there is still a significant income disparity, between the two genders,
and it was substantially worse in the historical past.

He who acquires a wife gets his best possession

wives are possessions

See equivocation.

When discussing a headstrong daughter, he is incredibly obscene:

When discussing a morally loose daughter, he is incredibly poetic.

Sirach is quite ambiguous actually, in the Greek and the Hebrew original as well. Looking at differing translations alone will give you some idea of the variety of interpretations. We should be wary of jumping to conclusions from translations based solely or primarily on the Septuagint, which is itself a translation of the Hebrew in a genre that is very difficult to translate (proverbial literature).

Sirach 42:13-14 reads in the Hebrew original (this is from the Dead sea scrolls):

(42:13) :3r12 כי מבגד יצא עש [[ ]] ומאשה רעת אשה׃
(42:14) :4r12 מטוב רוע איש מטיב אשה [[ ]] ובית מחרפת תביע אשה׃ >טוב רע איש

מחפרת מטוב אשה ובת מחרפת תביע חרפה<

Verse 13 is clear, but verse 14 is quite cryptic.

This is interpreted by Benjamin Parker and Martin Abegg as:

For from a garment comes a moth, and from a woman, woman's wickedness.

Better is the evil of a man than to be treated well by a woman. [[]] And a woman pours out a houseful of reproach. OR (this is another rendering) Better to be harmed by a man than to be shamed by a woman's goodness.

As for the interpretation of these verses, in regard to verse 13, I think this is as much saying that the wickedness of women frequently is sparked in the context of other women (gossip). As for verse 14, the renderings here are split by the ambiguity of the Hebrew word for "shaming." It could be saying many things, far more than what the Greek Septuagint suggests it could. One thing it could mean is that the so-called goodness or air quote goodness ("goodness") is worse than the badness/wickedness of men. That is as much to say, there is nothing so dangerous to a man (Sirach was written by a man to young men) as a woman that smiles and charms and means nothing good to that man.

Verse 14 could also mean (because the Hebrew word "ra'ah cannot only mean moral evil but also bad fortune) that the harm or bad situation that another man can bring another man into does not compare to the harm that a smiling, "good" woman can bring to him. I think these interpretations are the fairest to the original because Sirach has many good things to say about good women in another place.

As for the Sirach 25 verse, this does nothing more than restate what Protestants, Catholics, and Orthodox all affirm is Scripture (1 Timothy 2):

Ἀδὰμ γὰρ πρῶτος ἐπλάσθη, εἶτα Εὕα: καὶ Ἀδὰμ οὐκ ἠπατήθη, ἡ δὲ γυνὴ ἐξαπατηθεῖσα ἐν παραβάσει γέγονεν. σωθήσεται δὲ διὰ τῆς τεκνογονίας, ἐὰν μείνωσιν ἐν πίστει καὶ ἀγάπῃ καὶ ἁγιασμῷ μετὰ σωφροσύνης.

For Adam was first made, then Eve. And Adam was not deceived, but the woman was beguiled in her transgression. But she will be saved through childbearing, if they remain in belief and love and holiness with self-control.

Which is harder to accept in a modern context? If you're going to inveigh in this way against Sirach, then one must inveigh as well against Paul in 1 Timothy.

In Sirach 22:3, unfortunately, we have no surviving Hebrew manuscript available, so we have to rely on the Greek (and I suppose the Peshitta Syriac), which I can't find.
Here's the Greek:

αἰσχύνη πατρὸς ἐν γεννήσει ἀπαιδεύτου, θυγάτηρ δὲ ἐπ᾿ ἐλαττώσει γίνεται.

The shame of a father is in the generation of an untrainable child, but/and a[n] [untrainable] daughter at a loss is born.

Now the key issue is whether or not the word 'untrainable' is assumed to be present in meaning in the second half of the clause. This seems to be quite possible, because this type of thing often occurs in Hebrew poetry, particularly in the Psalms, where an adjective found in the first half of the clause is assumed to be functioning for a noun in the second half of the clause. Sirach says good things about good women elsewhere, so it does not make sense to be that the birth of a daughter in his eyes must necessarily be a loss, point blank. Also, we have a similar statement in Ecclesiastes 7 that both Protestants and Catholics have to recognize.

אֲשֶׁר עוֹד-בִּקְשָׁה נַפְשִׁי, וְלֹא מָצָאתִי: אָדָם אֶחָד מֵאֶלֶף, מָצָאתִי--וְאִשָּׁה בְכָל-אֵלֶּה, לֹא מָצָאתִי

The other verses do not prove too problematic to me. Sirach 36:29 is really typical of the way the Old Testament in general talks about women. Must that necessarily carry over into the new covenant? No. The last verse, Sirach 26:12, does not seem an issue either unless you're going to reject the book of Ezekiel, which has two sections that are often described as "p**nographic" and "not to be read in church" about midway through the book.

Finally, much of this book is written in the spirit of exaggeration and hyperbole. Things that contradict one another are often found. We have to be very careful when we read any isolated verse that we pull from the wisdom literature and have to be willing to read both sympathetically and in the context of the times.