: Slow buildup vs sudden introduction Let's say you're writing super hero fiction. The hero of your story is struggling against the villain, but suddenly awakens a power that allows him to best

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I'd introduce the build up first.
The appearance of a DEM is difficult to overcome with subsequent explanations, IMO without a hint of what is happening, this taints your story, especially if the guy with power is not that surprised and seems to know they were capable of this all along (as will be the case if the power is shown to be sought, later).
One exception is well-known comic-book heroes (or similarly well-known fictions, e.g. Transformers), so the reader knows what to expect from the start; we would not be surprised if a pre-Batman Bruce Wayne showed extraordinary inventiveness or fearlessness in battle.
Another exception is settings that allow for constantly new things: Magical or SciFi. It does not seem a DEM if some character in Lord of the Rings shows a magical ability for remote viewing or starting a fire. It does not seem a DEM if a brand new Star Trek being has the ability to divide itself into multiple beings, or magically transport a space ship to another galaxy.
But your setting does not sound like such a place, and your character is not already famously established with your audience, so I don't think these exceptions apply.
I would show some small hint that this power is possible; some small manifestation, then it is plausible that under duress, injured, bleeding and about to be killed, your character can summon something a thousand or million times greater than what he was doing for fun. Instead of struggling to light a candle with his mind and only succeeding after several sputtering starts, in his desperation in battle he incinerates the villain as if in a roaring furnace.
Or whatever your particular power is.

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The "Build up to it" route is the conventional way to tell the story. The progression is more or less linear. The protagonist's struggle to unlock the power is bound together with his struggle against the villain.

The "Introduce suddenly, explain later" route offers you several interesting, not mutually exclusive possibilities.

If your protagonist goes into a fight against the villain knowing that he does not have the power to defeat him, that in fact this is a doomed fight, the moment of finding the new power can be framed as a moment of eucatastrophe, of grace.
Once the new power has been found, you have on your hands a mystery: how to use it consistently, why it showed up now, etc.

The "mystery" aspect plays an interesting role in Naomi Novik's Uprooted: about a quarter of the way into the story, the MC does something which is considered impossible, though she doesn't know it at the time. This later leads to a re-examination of what is and isn't possible, how and why.

An important aspect of "Introduce suddenly, explain later" is that you absolutely must explain later. Otherwise it's a Deus ex Machina.

What option you ultimately pick has to do with the themes of your story, what story you want to tell. Do you want to tie the search for the power with the struggle against the Big Bad? Choose the first. Do you want your story to be about re-examining what is considered possible? Choose the second. Is the hard choice connected to accepting the power? Choose the first. Is the choice related to now having the power? Choose the second.

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"Hero" is different from protagonist.

Hero appears from nowhere and without explanation defeats the villain: This hero is not your protagonist. He is likely a paragon, someone that your protagonist looks up to. Emotionally, the reader is following the protagonist who feels there is no hope and the hero has abandoned them, or who still has faith in the hero despite his defeat and sulking inside that temple.

Hero struggles to learn the nature of his power: He must understand its origin, overcome the fear that he cannot control it, that it may change him. Once he taps into this power, he cannot go back to who he was before, but he must do this to save the people he loves. This hero is your protagonist.

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The way you describe them they are not different
You mention that one of the options is about building up small steps, hinting at the fact that he will unlock something.
The other option is about having small steps, which are hard to see, that you are hinting at with him learning something.
These are the same things, you just seem to think that there is a difference depending on how obvious you want to make him taking the steps. But there are still steps. And the reader can still pick up on them. (By the way, him going into a secluded temple without any information about what he is doing there is not really a small hint - that's a big "He has a secret weapon that he will use in his final fight against the Big Bad Evil Guy.)
Pros and Cons are largely the same. In one case you are just trying to be explicit, for example so that you can use the training as a way to showcase his character and possibly his mentor/friends/... while in the "other" approach you are trying to be secretive about it so that you can have "A Big Reveal" in the fight.
Be careful about what you think is that second option - you already mention that it might seem like a Deus Ex Machina and a lot of your readers in a super hero fiction might feel that way, too. It's a sudden secret power that nobody knows anything about before and therefore nobody can say whether the hero really had this ability before or you just became lazy and wanted to have a way for your hero to win this encounter. Simply unlocking the power might feel like a Deus Ex Machina, too, depending on your execution, but there your readers have some explicit things that show the character has been trying to learn and use specifically this power for specifically this fight.
In general you will probably want to go with the first route as the other one might feel a bit cheaper. If it's well executed the second can be good, though something like that is more often seen in mystery or crime - think about a crime-solving detective who finds some evidence that is not explicitly shown to the reader/viewer, but is the missing little detail to solving the case.

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