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No, you cannot counter Dragonlord Dromoka with Voidmage Apprentice.
Dromoka cannot be countered at all, it doesn't matter what is trying to counter it.
Also "When ~ is turned face up, ..." is a triggered ability. So flipping a card face up is a special action that doesn't use the stack, and once the card is face up its ability triggers and is put on the stack.
If instead you were trying to counter something like Abrupt Decay that "can't be countered by spells or abilities." you still couldn't counter it because you would still be using an ability to try and counter it. The "by spells or abilities" rider is there on spells with targets so that if a spell has no legal targets it can still be countered by the game rules.
Is there a limit on Dromoka's can't be countered? Nope, so it can't be countered.
And the flip itself isn't an ability, but the triggered counter is an ability.
The answer to the question as written is no. But, the answer to the question in the subject line is different.
The answer is, it depends on your definition of "counter a spell."
If your definition is the usual, technical definition, then as other answers have pointed out, you cannot counter a spell that cannot be countered.
However, if your definition is a bit broader, such as, "stop that spell from resolving, no matter what." Then the answer is YES. Here is what we've found so far: