Measuring is clearly not the answer here. Is there some trick to prove it is $|1\rangle$ state?
No, no trick. You can’t prove it.
One way to think about this is what if you had either the 1 state, or something that is arbitrarily close to the 1 state with a tiny amount of 0. You’re essentially asking if it’s possible to perfectly distinguish them. But if you could, you could copy them, and you’d have perfect cloning of a pair of non-orthogonal states, and that’s impossible.
I assume that you mean to say that initially we are not aware of the quantum state of the qubit. The answer to your question is then, No.
Say, the qubit was in some state $\alpha |0\rangle + \beta |1\rangle$. Now, if by any way we are able to prove that $\alpha =0$ (or $\beta =0$, for that matter), with a single shot experiment, then in a sense we are able to 'know' the quantum state of the qubit and hence we can clone it. This is impossible.