Can any of the texts of the New Testament, other than Acts, 1 Thessalonians or 2 Peter (for which I already have evidence), be interpreted in such a way that they MUST have been written before the end of the first century?
One of the most important indicators of the date of a manuscript is the presence of any references to chronologically known events. In the case of the New Testament, a major event is the destruction of Jerusalem and the Temple in 70 AD/CE.
For example, the account in Acts about Paul's involvement and arrest over an incident at the Jerusalem Temple indicates that the Temple had not been destroyed yet. Someone challenging this reference would have to come up with some extraordinary contradicting evidence.
However, the absence of a reference to a major event, doesn't necessarily mean that a manuscript preceded the event, and this is where historical contexts are subject to argument--would the writer have been expected to mention the event in the context of the subject of the manuscript.
Allowing that a few years must pass between the writing of a document and its surfacing in quotations in various parts of the Roman world (no email or Amazon publishing back then), a quotation of a New Testament document by the Apostolic Fathers would be very strong evidence that the document in question was written in the 1st Century.
I will principally focus on quotations by 1 Clement (written no later than 96), Ignatius (~107), Polycarp (~107), with occasional reference to other documents as well.
Every book in the New Testament is quoted or paraphrased in the Apostolic Fathers except John, Philemon, 2 John, and Jude. The allusions to James, 2 Peter, 3 John, and Revelation are possible but not so strong. So that’s at least 19 books of the New Testament solidly dated to the 1st Century on quotations alone.
Matthew: Quoted by Ignatius & Polycarp. Eduard Masseux argued effectively that Matthew is also quoted in 1 Clement, given exact correspondence in unusual Greek wording. (The Influence of the Gospel of Saint Matthew on Christian Literature Before Saint Irenaeus: Book 1, The First Ecclesiastical Writers pp. 21-24) Matthew is also quoted by the Didache, which may have been written in the 1st century. A passage from Matthew is found in the Epistle of Barnabas; the date of this epistle is uncertain, but Robinson (Redating the New Testament Ch. 10) makes a compelling argument that it was written in the 1st Century. Both the Farrer Hypothesis & the Two Gospel Hypothesis claim Matthew was quoted by Luke. The Two Gospel Hypothesis claims that Matthew was quoted by Mark as well.
Mark: Quoted by Polycarp. The Two Source Hypothesis and the Farrer Hypothesis claim Mark was quoted by Matthew and Luke.
Luke: Quoted by Ignatius, Polycarp, and 1 Timothy. May be used by 1 Clement as well.
John: No clear surviving quotations from the Apostolic Fathers
Acts: Quoted by Clement, Ignatius, and Polycarp
Romans: Quoted by Clement, Ignatius, and Polycarp
1 Corinthians: Quoted by Clement, Ignatius, and Polycarp. Very explicitly discussed in 1 Clement
2 Corinthians: Quoted by Polycarp; inferred by 1 Clement
Galatians: Quoted by Polycarp
Ephesians: Quoted by Clement, Ignatius, and Polycarp
Philippians: Quoted by Polycarp
Colossians: Quoted by Ignatius
1 Thessalonians: Quoted by Ignatius & Polycarp
2 Thessalonians: Quoted by Polycarp
1 Timothy: Quoted by Polycarp; probably quoted by 1 Clement
2 Timothy: Quoted by Polycarp
Titus: Quoted by Clement
Philemon: No clear surviving quotations from the Apostolic Fathers
Hebrews: Quoted extensively by Clement; quoted by Polycarp
James: Allusion in 1 Clement
1 Peter: Quoted by Polycarp
2 Peter: Allusion in 1 Clement
1 John: Quoted by Polycarp
2 John: No clear surviving quotations from the Apostolic Fathers
3 John: Allusion in Polycarp
Jude: No clear surviving quotations from the Apostolic Fathers
Revelation: Possible allusion in 1 Clement
What about Philemon?
Although there are no known quotations of Philemon prior to Tertullian, Philemon is solidly attributed to Paul by both conservative and liberal scholars. Paul died sometime between the fire of Rome (July of 64) and the death of Nero (June of 68), putting Philemon firmly in the 1st Century.
P52 is the earliest known surviving manuscript of the New Testament. It was found in Egypt and dates to approximately the year 125. It would have taken some time for the Gospel of John to have made its way to Egypt, suggesting it was written well before that date.
Revelation is explicitly attested as a 1st century document by Irenaeus and implicitly by Justin as well. Irenaeus, as a student of one of John’s disciples, was in a good position to know who wrote it.
With his emphatic stance on relying on material handed down by the apostles, Irenaeus was unlikely to quote as scripture something that was written in his lifetime. This is not as solid a claim for 1st Century authorship as the quotations by Clement, Ignatius, and Polycarp, but it is a real problem for late dates of New Testament documents. Irenaeus quotes every book of the New Testament except Philemon, 2 Peter, 3 John, and Jude.
That gives us external evidence for every New Testament document dating to the 1st century (most of them definitely, some of them probably) except the epistle of Jude.
Never in the New Testament is the destruction of Jerusalem and its temple (AD 70) mentioned as something that has already happened. Eminent New Testament scholar John AT Robinson used this data, and a veritable treasure trove of inquiry into other relevant scholarship, to suggest that every New Testament document was likely written no later than 70. His seminal work on the subject is available online here.
Matthew & Hebrews provide arguments that don’t really work if the temple has been destroyed. (see Matthew 17: 24-27, Hebrews 10:2)
Several books of the New Testament, mostly especially Matthew, Hebrews, James, and Jude, are written to people who are simultaneously Jewish and Christian. This was common in the first generation after Easter and become increasingly uncommon thereafter. Judaism & Christianity decisively separated into distinct religions during the Flavian era (70s-90s), meaning the audience presupposed by these documents did not exist in the 2nd Century, and they should thus be dated to the 1st.
A deep dive on the Gospels
(This is the boring part you can skip) I am in the process of creating a series on the writing of the Gospels, which has bearing on this question. It firmly puts the writing of the Gospels in the 1st Century. If you are interested in my research, you can find it here.
There will probably always be skeptics with an axe to grind, but if we stick with the evidence, we can pretty solidly claim that most, if not all, of the books of the New Testament were written in the 1st Century.