How does lifelink affect indestructible creatures?
You will gain 4 life.

Lifelink and indestructible have no special interactions with each other. Your 4/4 will deal 4 damage to the 1/1, and you will gain 4 life. This is the same as if your 4/4 had dealt damage to any other creature. It doesn't matter how much toughness the defending creature has, or whether it will eventually die.


  510.1a Each attacking creature and each blocking creature assigns combat damage equal to its power. Creatures that would assign 0 or less damage this way don’t assign combat damage at all.