According to Catholicism, will those who are alive on the Day of Judgement die physically? At  1 Thessalonians 4:15–17 (NRSVCE) we read:


  For this we declare to you by the word of the Lord, that we who are alive, who are left until the coming of the Lord, will by no means precede those who have died.  For the Lord himself, with a cry of command, with the archangel’s call and with the sound of God’s trumpet, will descend from heaven, and the dead in Christ will rise first.   Then we who are alive, who are left, will be caught up in the clouds together with them to meet the Lord in the air; and so we will be with the Lord forever.


Do the words of St. Paul imply that those who are alive on the Day of Judgement will not face physical death, but will directly enter Judgement?  What do the Catholic Church's teachings say about such a prospect?
Short Answer
Yes, the people alive on the last judgment will go, body and soul, into their final destination, and thus not experience a separation of the body and the soul, i.e. death.
Long Answer
What is death?
Death is understood in the Catholic faith as a separation of the soul and the body. As Father John Echert writes:

At the moment of death, the soul is separated from the body and no
longer sustains order within the natural body

What happens upon death?
When someone dies, they undergo particular judgment defined by the Catholic Encyclopedia as:

The Catholic doctrine of the particular judgment is this: that
immediately after death the eternal destiny of each separated soul is
decided by the just judgment of God. Although there has been no formal
definition on this point, the dogma is clearly implied in the Union
Decree of Eugene IV (1439), which declares that souls leaving their
bodies in a state of grace, but in need of purification are cleansed
in Purgatory, whereas souls that are perfectly pure are at once
admitted to the beatific vision of the Godhead (ipsum Deum unum et
trinum) and those who depart in actual mortal sin, or merely with
original sin, are at once consigned to eternal punishment, the quality
of which corresponds to their sin (paenis tamen disparibus). The
doctrine is also in the profession of faith of Michael Palaeologus in
1274, in the Bull &quot;Benedictus Deus&quot; of Benedict XII, in 1336, and in
the professions of faith of Gregory XIII and Benedict XIV.

After the particular judgment, their soul will be in the condition of eternal suffering (Hell), eternal happiness (Heaven), or a purging period for temporal punishment due to sin (Purgatory) then eternal happiness. Eternal happiness is eventually guaranteed when a soul reaches purgatory since no soul in purgatory ends up in Hell according the the Catechism of the Catholic Church:

1031 The Church gives the name Purgatory to this final purification of
the elect, which is entirely different from the punishment of the
damned. The Church formulated her doctrine of faith on Purgatory
especially at the Councils of Florence and Trent. The tradition of the
Church, by reference to certain texts of Scripture, speaks of a
cleansing fire
As for certain lesser faults, we must believe that, before the Final
Judgment, there is a purifying fire. He who is truth says that whoever
utters blasphemy against the Holy Spirit will be pardoned neither in
this age nor in the age to come. From this sentence we understand that
certain offenses can be forgiven in this age, but certain others in
the age to come.

Once a person faces particular judgment, there is no changing that judgment as Father Paul K. Raftery writes:

The state of the dead up to this point [after particular judgement] had been one of souls abiding
in separation from the body. But at the general judgment this
separation of body and soul will be for ever ended. From this moment
on men will have that union of body and soul God always intended for
them. However, the appearance of the body for these two groups will be
vastly different. The bodies of the blessed will be radiant with
divine light, but those of the condemned will manifest the twisted and
dark soul that is within (See Suppl. 75, 2, ad 3). Whatever the state
of the soul is, that the body will reveal.

What happens at the last judgment?
At the last judgment, the souls of the dead will be reunited with their bodies, and the consequences of their actions will be made known to all according to the Catechism paragraph 1038:

1038 The resurrection of all the dead, &quot;of both the just and the
unjust,&quot; will precede the Last Judgment. This will be &quot;the hour
when all who are in the tombs will hear [the Son of man's] voice and
come forth, those who have done good, to the resurrection of life, and
those who have done evil, to the resurrection of judgment.&quot; Then
Christ will come &quot;in his glory, and all the angels with him. . . .
Before him will be gathered all the nations, and he will separate them
one from another as a shepherd separates the sheep from the goats, and
he will place the sheep at his right hand, but the goats at the left.
. . . And they will go away into eternal punishment, but the righteous
into eternal life.&quot;

and paragraph 678 respectively:

678 Following in the steps of the prophets and John the Baptist, Jesus
announced the judgment of the Last Day in his preaching. Then will
the conduct of each one and the secrets of hearts be brought to
light. Then will the culpable unbelief that counted the offer of
God's grace as nothing be condemned. Our attitude to our neighbor
will disclose acceptance or refusal of grace and divine love. On
the Last Day Jesus will say: &quot;Truly I say to you, as you did it to one
of the least of these my brethren, you did it to me.&quot;

So how does the last judgment apply to those who are still living?
The same happens for the living at the time, and they will NOT die (and thus, strictly speaking, not face particular judgment), but rather directly undergo the last judgment and be sent to their final condition, body and soul, for eternity.
The parameters that apply to both the living and the dead, however, will be identical, save that those living at the time of the last judgment will not need to have their soul reunited with their body.
And thus we go full circle. Your reading from 1 Thessalonians confirms this:

Then we who are alive, who are left, will be caught up together with
them in the clouds to meet the Lord in the air, and so we will always
be with the Lord.