"Push buttons" action is unlocked once you have an enemy sim. To make an enemy Sim, you have to reach level 4 in Mean relationship and perform "Declare disdain" action.
Find a Sim you dislike and perform Mean actions against him/her. Once the relationship level reaches 2, you can start "Declare rivalry" social event that changes status from Acquaintance to Rival.
While the relationship status is Rival, keep doing actions with him/her in Mean manner to reach level 4 to unlock Declare disdain social event to change status from Rival to Enemy.
After the social event, you become an enemy against the Sim and Push buttons action is unlocked.
Now you can perform Push button of an enemy to accomplish the Feud Starter aspiration. It just takes few seconds.
The Sims Mobile Player Level
Aging and Retirement in The Sims Mobile
The Sims Mobile Friend Request forum
Party in The Sims Mobile
Aging and Death