If the correlation is positive, can I conclude that one variable predicts another variable?

Yes.

I see, it is unclear for the predictor-outcome relationship in a correlational analysis.

No. A positive correlation only shows that the two variables are correlated positively. However, when we say A predicts B, we refer to a causal relationship between A and B, that is, A is the cause of B. Evidence supporting this causal statement should include: 1) Varying together. 2) No alternative explanations for the covariance. 3) A happens before B. Therefore, the evidence of a positive correlation is not enough to claim a causal relationship.