Does Karthus get to repeat Deathknell abilities if he dies at the same time?

Yes, if Karthus, Eternal dies at the same time as another unit with a Deathknell ability, that unit's Deathknell will still trigger an additional time.

This interaction occurs because Karthus’s ability ("Your [Deathknell] effects trigger an additional time") is a passive ability, not a triggered ability. According to RiftJudge FAQ #5353 and FAQ #9317, passive abilities do not need to evaluate their own trigger condition at the moment they die; they apply to the game state as long as the card is on the board.

Why this works:

1. Passive Ability: Because Karthus's ability is passive, it remains active at the exact moment he is killed. It does not need to be "alive" to evaluate whether it should be active; it simply modifies the rules of the game for as long as he is on the battlefield.
2. Simultaneous Events: When Karthus and another unit with a Deathknell trigger are both sent to the trash simultaneously (e.g., via lethal combat damage or a board wipe), Karthus's passive effect is still being applied to the game state when the other unit's Deathknell is triggered.
3. Result: The other unit’s Deathknell triggers once for its own death, plus one additional time due to Karthus’s passive ability.

This is distinct from triggered abilities (like those of Viktor, Leader), which cannot trigger if the unit leaves the board at the same time the trigger condition is met, as those abilities must evaluate their own trigger condition at that specific moment. Since Karthus is a passive effect, he successfully modifies the Deathknell trigger of the other unit as it occurs.