Arjuna wasn't enlightened, of the many characters of the Mahābhārata, only two were enlightened: Kṛṣṇa, and then Balarāma. The two of course represent divergent attitudes towards enlightenment-in-general. For Kṛṣṇa, even though he was “enlightened,” he involved himself in the politics, war, and romance of day-to-day life (this was, by his own admission, for the purpose of effecting justice throughout the world). For Balarāma, there was almost no such involvement. It's for this reason that Balarāma/Saṅkarṣaṇa is sometimes regarded as Śiva-like, whereas Kṛṣṇa is regarded as Viṣṇu-like.
Arjuna was just a particular jiva who happened to be friends with a particularly-enlightened person (i.e. Kṛṣṇa). Unfortunately being friends with an enlightened person does not necessarily mean that you yourself will become enlightened.
Enlightenment, if we are to talk about it, is more of a solo endeavor that requires great grit, determination, and bravery to actually bring about. So it's not really a random thing, consequently it is very rare (regardless of the time that you live in). Shockingly, it is thus found that only two characters in the great Epic are actually seen to be of significance (i.e. Vāsudeva and Saṅkarṣaṇa). It is for this reason that the Jainas probably choose to focus on Baladeva and Vāsudeva alone, to the exclusion of the other characters.
(It is also probably for this reason that the Epic and the Purāṇas themselves seem to depict Baladeva and Vāsudeva as co-incarnations of Viṣṇu. Not even Vyāsa was enlightened.)
This notion that enlightenment/salvation is rare, is not really unique to Hinduism. The Shīʿa Imāms make a similar confession, in a well-known narration involving Jaʿfar al-Ṣādiq, he implicitly states to the narrator that there are less than 17 believers on Earth. (via Kitāb al-Kafī, ch. 96, no. 4)
According to the pseudonymous Jed McKenna, whose philosophy is almost identical to that of Śaṅkarā's Advaita Vedānta, there are only around 12 enlightened people on Earth, some of whom he doesn't know. (via Jed Talks #1, p. 53)
The point is that enlightenment/salvation/belief is rare. Of the manifold characters of the Mahābhārata, Kṛṣṇa achieved this state, and perhaps Balarāma. Arjuna was just Kṛṣṇa's worldly friend. There is no doubt that Kṛṣṇa's and Arjuna's friendship was deep and affectionate, but their respective levels were not comparable, and so it is for this reason that Arjuna was able to learn a little bit (at least temporarily) through osmosis, especially via the philosophy that Kṛṣṇa preached right before the war, but ultimately, none of it stuck.