What happens if Elder Dragon enters the field and there is a dmg unit?

Yes, that damaged enemy unit will die the moment Elder Dragon enters the field.

Elder Dragon's passive ability — "Any amount of your damage is enough to kill enemy units" — redefines what counts as lethal damage for enemy units that have any of your damage marked on them. As soon as Elder Dragon hits the board, the game performs a state check. If an enemy unit already has even 1 damage from you (from a previous attack, spell, etc.), that damage is now considered lethal, and the unit is killed immediately.

Two important limits:

• This only applies to your damage on enemy units. Damage dealt by your opponent to their own units won't kill them.
• This happens before reactions can be played to Elder Dragon's "When you play me" triggered ability, so there's no window to prevent the kill unless you had a pre-existing replacement effect (like Counter Strike) ready.