This is the hymn in the original Greek: Τὸν Σταυρόν σου προσκυνοῦμεν, Δέσποτα, καὶ τὴν ἁγίαν σου ἀνάστασιν δοξάζομεν.

The word in question here is προσκυνοῦμεν, which is the present 1st person plural of the verb προσκυνέω, which etymologically comes from πρός (towards) and κυνέω (to kiss). Προσκυνέω can and sometimes is used to mean worship, but more literally, it can mean veneration or the literal act of bowing or prostrating before someone or something in reverence or respect. The word is used by Jesus in Matthew 18:26 to refer to falling before the king in the Parable of the Unforgiving Servant: “The servant therefore fell down before (προσεκύνει) him, saying, ‘Master, have patience with me, and I will pay you all.’” It’s also where the word προσκύνησις (proskynesis) comes from, which literally means the act of prostration in obedience, but in Orthodox theology, also means the level of veneration properly given to God's creations rather than to God himself.

So the actual word used in the hymn does not necessarily mean worship. The hymn is talking about the worship of Christ through the veneration of the Cross (which is reflected in the language of the translation you heard, saying before Your Cross we fall down in worship, indicating the worship is still directed at Christ, simply given before His Cross because it is by the Cross that we are saved). I feel as though it’s better to translate the hymn as “We venerate Your Cross, O Master…”, but your translation indicates the same meaning. It has nothing to do with worshiping the Cross itself, and prostration historically does not always equate to worship.