If I play a unit to Star Spring as an ambush reaction to my opponent playing a spell to deal enough damage to kill another unit at a battlefield, can I then use Star Spring's ability to push that unit back and save it?

To answer your question: Whether you can save the unit depends on the specific targeting requirements of the damaging spell.

While playing an Ambush unit to Star Spring triggers its ability, moving the unit to your base does not automatically save it from all damaging spells.

The Reasoning

1. Targeting Restrictions: According to FAQ #1794 and FAQ #5858, if a spell specifically targets a unit "at a battlefield," moving that unit to your base makes it an invalid target, causing the spell to fizzle. However, if the spell does not have a battlefield restriction, FAQ #1465 states that moving the unit to your base does not make it an illegal target, and the spell will still resolve and deal damage.
2. Chain Resolution: When you play your Ambush unit, Star Spring's triggered ability is placed on the chain. You can resolve this ability to move your other unit to your base. Because the damaging spell is already on the chain, it will resolve after your triggered ability finishes.
3. Effect of Movement: If the spell requires the unit to be at a battlefield, the movement will save the unit. If the spell does not have this restriction, the unit will still take the damage and die.

Sequence of Events

1. Opponent's Action: Opponent plays a damaging spell targeting your unit. This spell is now on the chain.
2. Your Reaction: You play an Ambush unit to Star Spring. This triggers Star Spring's ability: "The first time a player plays a non-token unit here each turn, they may move another unit they control here to its base."
3. Chain State: The chain now contains (from bottom to top): [Damaging Spell] -> [Star Spring Trigger].
4. Resolution:
   • The Star Spring trigger resolves first (LIFO). You move your unit to your base.
   • The Damaging Spell then resolves. If the spell requires the unit to be at a battlefield, it fizzles. If it does not, the unit is still a legal target and takes the damage.

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