From Dane Ortlund:
Seems to me that while it need not be the main point of every NT book, nevertheless every NT book in some way fulfills the hope of the OT, though each from its own perspective. One former prof of mine used to say that the NT is a 27-volume commentary on the OT. Truth to that.
Matthew fulfills the OT’s hope for a Messiah, a Christ, an anointed son of David who would save God’s people (1:21).
Mark fulfills the OT’s hope for a coming Son of God who would inaugurate God’s kingdom (1:1, 14–15).
Luke fulfills the OT’s longing for God to come and set right the world’s injustices—reversing rich and poor, oppressors and oppressed, satisfied and hungry, outsider and insider (19:10).
John fulfills the OT’s longing for the tabernacle/temple to do decisively what it was always meant to do—unite God and man in restored fellowship (1:14; 2:21; 14:6).
Acts fulfills the OT by bringing God’s mercy to the nations (1:8; 9:15).
Romans fulfills the OT by showing the supreme manifestation of the righteousness of God, in Jesus, bringing resolution to the constant OT tension between God’s justice and his mercy (1:17; 3:21–26).
1 Corinthians fulfills the OT by showing, in Christ, the climactic way in which God destroys the wisdom of the wise (1:19).
2 Corinthians fulfills the OT’s repeated pattern of strength through weakness (12:9–10), supremely in Christ (13:4), in whom all the promises of God are clinched (1:20).
Galatians fulfills the OT by showing that Jesus’ atoning work (3:13) at just the right time (4:4–5) is the reason that the real children of Abraham are those who are of faith (3:7–9).
Ephesians fulfills the OT by revealing the “mystery” long hidden—that Christ, by virtue of his death and resurrection, unites Jews and Gentiles in one renewed people of God (3:5–6).
Philippians fulfills the OT by showing that the church is the real circumcision (3:2–3).
Colossians fulfills the OT by showing that another Adam, likewise the image of God (1:15), has fulfilled the creation mandate of Genesis 1:28 to bear fruit and increase, so that we who are united to this second Adam can now do what the first Adam failed to do—bearing fruit and multiplying (1:10).
1 and 2 Thessalonians fulfill the OT’s hope of judgment on God’s enemies by showing that Jesus received this judgment, so that God’s punitive judgment, which is surely coming, now will fall only on those who reject Jesus (1 Thess 5:1–10; 2 Thess 1:5–12).
1 and 2 Timothy fulfill the OT by showing that the true warfare of God’s people is not against the Amalekites and Amorites and others but against sin and Satan (1 Tim 1:18; 6:12; 2 Tim 2:3–4), a war that cannot be lost because of the Savior anticipated in the OT (2 Tim 3:15).
Titus fulfills the OT’s underachieved efforts to redeem a people for God who are his own possession, zealous for good works (2:11–14).
Philemon fulfills the OT’s insistence that love be from the heart (v. 14).
Hebrews fulfills the OT’s longing for a perfect priest and final sacrifice to usher in the new covenant (8:1–13).
James fulfills the OT’s call for obedience to the law by showing that such obedience is fulfilled in one thing—active love (1:12; 2:8–26).
1 and 2 Peter fulfill the OT’s calling to Israel to be a royal priesthood and a holy nation (1 Pet 1:4–12)—a corporate fulfillment that happens only because of another fulfillment that is not only corporate but also individual, this time of Isaiah 52–53 (1 Pet 2:22–25).
1, 2, and 3 John fulfill the OT by showing that through Christ we are once more, like Adam, sons of God, and now able to fulfill the OT law through love (1 John 3:1 and passim).
Jude fulfills the exodus in the OT by showing that ultimately is was Jesus who provided this rescue (Jude 5; cf. 1 Cor 10:4).
Revelation fulfills the OT by showing that Jesus has conquered our great enemy, death, which was introduced in Eden (Rev 1:18; 21:4).