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Suppose that a qubit in an initial state that we don't know was measured, and the result was 1. Is it possible to know the initial state of the qubit by the result measured? And if the result would have been 0?

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No. The only thing you know is that $\langle \psi | 1\rangle\neq 0$.

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I have updated the question. – João Reis Jan 15 at 14:42
Sorry, but there's no physical difference between 0 and 1, they are just labels. So the answer is the same. You would know that $\langle\psi|0\rangle\neq0$. This is the interesting thing about quantum mechanics. A measurement usually can't tell you everything about the state. If you have any doubt I can explain you some details. – Bzazz Jan 15 at 14:46
I was just thinking about that, that the only thing we would know is that $\langle \psi | 0\rangle\neq 0$ too. Thank you! – João Reis Jan 15 at 14:50
You're welcome. – Bzazz Jan 15 at 14:51
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If you've got a prior distribution for the qubit's initial state, doesn't the measurement give you additional information that you can use to update your prior using Bayes? So although you don't know everything about the initial state, you know a little more about it than you did before. – EnergyNumbers Jan 15 at 15:01

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