Steve Rogers was worthy to wield Mjolnir long before he became Avengers: Age of Ultron, that had seemed to suggest Captain America was worthy. That naturally raised the question of whether he'd recently become worthy, or whether he had been all along.
In the MCU, to be "worthy" is to be willing to stand up for others - no matter the personal cost. In the first Thor: The Dark World, when the Convergence allowed the God of Thunder's defense of the Nine Realms to be seen by all its peoples, proving he was ready to become King of Asgard.
If this is the case, then Steve Rogers was worthy long before he ever became Captain America, which is clearly revealed in Captain America: The First Avenger. Abraham Erskine chose him to become America's first super-soldier because he saw qualities beyond the physical, a greatness of spirit in a fragile frame. He demonstrated this to Colonel Phillips by tossing what seemed to be a live grenade to the floor in front of Steve Rogers and his fellow soldiers; where the others scattered, Rogers covered it with his own body, instinctively willing to sacrifice himself for the sake of those around him. This is all the more remarkable when considering the other soldiers included at least one man Erskine described as a bully, who would undoubtedly have been happy to prey upon the weakling Steve Rogers. Yet, Rogers was unhesitating in his willingness to die to save their lives.
Abraham Erskine wasn't looking for a physical powerhouse; he had a super-soldier serum that could grant a man power, after all. Rather, he was looking for a good man, and seemed to have had pretty much the same criteria as Mjolnir. To Erskine, and to Mjolnir, a good man is someone who will stand up for others; to be worthy is to be heroically self-sacrificial, instinctively willing to give up your life. Steve Rogers proved himself worthy when he unthinkingly demonstrated his character that day; he was worthy enough to become Captain America, and worthy to wield the power of Thor.
Looking forward, it will be fascinating to see how others come to prove themselves worthy. Although Mjolnir was destroyed in Thor: Love & Thunder, and Marvel is mirroring the comics in granting her, too, the power of Thor. In the comics, Jane Foster was worthy for the same reason as the Odinson and Steve Rogers in the MCU. She believed the world needed a Thor, and was willing to become that hero even though the transformations reversed the chemotherapy she was undergoing to save her life. It will be fascinating to see how this plays out in the MCU.
