Step 1: Fix the anode and cathode first.
In this Zn-Cu cell, zinc gets oxidised at the anode, $\text{Zn} \to \text{Zn}^{2+} + 2e^-$, and copper gets reduced at the cathode, $\text{Cu}^{2+} + 2e^- \to \text{Cu}$.
Step 2: Track electrons before converting to current.
Electrons physically leave the zinc anode and travel through the external wire toward the copper cathode.
Step 3: Flip direction for conventional current.
Conventional current runs opposite to electron flow, so externally it travels from Cu to Zn, exactly matching Statement I, so that part is correct.
Step 4: Check the mass changes.
Zn keeps dissolving away, so its mass drops, while Cu keeps depositing fresh metal, so its mass rises, the reverse of what Statement II claims.
Final answer: Option 3, Statement I is correct and Statement II is not correct.