Yes, this seems to be a common way that it was used. As another answer pointed out, the noun is not found elsewhere in the New Testament. However, Luke was familiar with (arguably, an imitator of) both LXX and Classical Greek, and there are multiple examples of ἀγωνία with this sense available there. Because context is required, I have included only English translations here with links to the Greek to avoid an excessively long post.
2 Maccabees 3 shows the emotional aspect of ἀγωνία well. The English quoted here is from NETS; at the link you can see the Greek and Brenton. Verses 13-16:
But the other one, because of the commands he had from the king, said that this money must in any case be confiscated for the king’s treasury. So he set a day and went in to direct the inspection of these funds. There was no little distress (ἀγωνία) throughout the whole city. The priests prostrated themselves before the altar in their priestly vestments and called toward heaven upon him who had given the law about deposits, that he should keep them safe for those who had deposited them. To see the appearance of the high priest was to be wounded at heart, for his face and the change in his color disclosed the anguish (ἀγωνία) of his soul. For terror and bodily trembling had come over the man, which plainly showed to those who looked at him the pain lodged in his heart.
Another example, from Aristotle, On breath:
But there is evidence that pulsation has no connexion with breathing; for whether a man breathes rapidly or evenly, heavily or quietly, the pulse remains the same and unaltered, but irregularity and excitement occur during some bodily ailments and in conditions of fear, expectation, and conflict (ἀγωνία) in the soul.
This translation brings out the point made by another answer that the basic meaning has to do with conflict. In the Aristotle passage as well as the others quoted here, this has been extended to include intra-personal, psychic conflict. To my ear, this stretches the sense of the term in English,1 but the context makes the meaning clear.
One more example, from Demosthenes, Speeches:
He was so nervous, and so much worried by the fear (ἐν φόβῳ καὶ πολλῇ ἀγωνίᾳ) that, in spite of his Thracian success, his enterprise would slip from his fingers if you should intervene before the Phocians perished.
Anxiety and fear are very much part of the context in each of these examples. This is mentioned in definition 3 of LSJ and is the primary sense of both the noun and the verb as described in BDAG (in contrast to ἀγών.) Both of the lexicons include additional references that could be tracked down.
1. Most likely because the English has been corrupted beyond recognition by psychoanalytic constructs regarding “intra-personal conflict”.
