So what you do know automatically is that Cu must be oxidized to Cu+, because this is a redox reaction, and Cu has no other option.
Cu ---> Cu+
From there, you now know something else needs to be reduced (lower in oxidation number). But you have to decide if the N will be reduced (+5 --> +2) or H will be reduced (+1 --> 0).
The only way you can decide between this is by knowing which is more easily reduced. In this case, HNO3 --> NO has a positive reduction potential, while H --> H2 has a 0 reduction potential. It's also helpful to think that Nitrogen is more electronegative than Hydrogen, therefore, N is more likely to be reduced (because electronegative = likes electrons).