What is the earliest systematic New Testament commentary? Were Origen's commentaries the earliest attempt to systematically exegete the New Testament texts, so far as there is any record?
Origen's commentaries do indeed appear to be the earliest surviving line-by-line, self-contained biblical commentaries.  This becomes evident when we examine his closest competitors.  We'll look first at Greek-speaking authors, and then at Latin.
Greek
About a hundred years before Origen's commentaries were written, Papias of Hierapolis wrote a lengthy work entitled Exposition of the Sayings of the Lord.  Sadly, virtually all of it has been lost, making its contents a matter of some scholarly dispute.  It's likely, though, that the work was primarily a setting down of the oral tradition that Papias knew, and not a systematic exegesis of one or more of the gospels.
Another prolific 2nd-century Greek author was Irenaeus, and in his work Against Heresies he regularly appeals to Scripture to point out the errors of his opponents.  Particularly in Book 4 and Book 5 he focuses his efforts on interpreting the sayings of Jesus and the epistles of Paul, but again, not in a systematic fashion.
Next we turn to the School of Alexandria, out of which came Origen.  Two masters preceded him, Pantænus and Clement of Alexandria.  Of the former no works survive, and none of Clement's surviving works meet our criteria.  But Eusebius speaks of eight volumes by him, called Outlines, that may have been commentaries:

There are another eight volumes of his Outlines, in which he names Pantaenus as his teacher and explains his interpretations of Scripture as well as the traditional (6.13.2, Maier translation)
In the Outlines he has given brief explanations of all the canonical Scriptures, including even the disputed writings. (6.14.1)

But since none of these exist today, we are left to conclude that Origen's commentaries are the earliest we have in Greek.
Latin
Paul J. Griffiths helpfully summarizes the early history of commentaries in Latin:

The first surviving self-contained commentaries on biblical books to have been composed in Latin were by Victorinus of Petrovium (d. 304), who was active in what is today Slovenia [...].  A little earlier in Roman Africa, portions of various polemical works by Tertullian were composed in commentarial form (for example, the fourth book of the Adversus Marcionem, on Luke [...]), even though Tertullian composed no works whose sole purpose is commentary on a particular biblical book. (Religious Reading, 151)

Victorinus of Petrovium died fifty years after Origen, but Tertullian was Origen's contemporary.  And indeed, in Book 4 of Against Marcion, Tertullian works through the Gospel of Luke systematically but polemically.  He wrote this about AD 208, which is around the time that Origen began to produce his commentaries.
Summary
The evidence outlined here indeed suggests that Origen's commentaries are the earliest surviving systematic and self-contained examples of New Testament commentaries.  Earlier commentaries may have been written, but have since been lost.  And though Tertullian wrote what we might fairly consider commentaries, such as one of the Gospel of Luke, they formed parts of his polemical writings, and in any case are dated to roughly the same time as Origen's.