If an Elder Dragon is played and pings one of my units for 1 damage to kill it, do I have time at any point to play Janna to that area to heal the unit and stop it from dying?

To answer your question: No, healing your unit with Janna will not stop it from being killed by the Elder Dragon's play effect.

While you have the opportunity to play Janna in response to the Elder Dragon's triggered play effect, healing the unit does not prevent its death.

Reasoning

The interaction relies on the distinction between the Elder Dragon's passive ability and its triggered play effect:

1. The Triggered Ability: When you play Elder Dragon, its ability to deal damage to a unit is placed on the chain.
2. The Passive Ability: The Elder Dragon’s passive ability states that any amount of damage dealt by its controller is enough to kill an enemy unit. This modifies the definition of lethal damage for any unit that has damage marked on it by the Elder Dragon's controller.
3. The Resolution:
   • You play Janna in response to the Elder Dragon's triggered ability.
   • Janna resolves first: She heals your unit, removing any pre-existing damage.
   • The Elder Dragon's triggered ability then resolves: It deals 1 damage to your unit.
   • Because the Elder Dragon’s controller is the source of this damage, the passive ability immediately recognizes that the unit now has damage marked on it.
   • Per the modified definition of lethal damage, that 1 damage is sufficient to kill the unit.

Healing the unit before the damage is dealt does not protect it because the damage event itself creates the condition—marked damage from the Elder Dragon's controller—that triggers the modified lethal threshold.

Sequence of Events

1. Play Phase: Opponent plays Elder Dragon. Its triggered ability is placed on the chain, targeting your unit.
2. Reaction Window: You play Janna in response to the Elder Dragon trigger.
3. Resolution:
   • Janna resolves: Your unit is healed.
   • Elder Dragon trigger resolves: Your unit takes 1 damage.
4. State Check: Because the unit now has damage marked on it by the Elder Dragon's controller, the passive ability applies. Per the rules regarding passive lethal modification, since the unit has non-zero damage that is considered lethal, it is killed.

Note: I could not fully verify this ruling against the retrieved rules.