Step 1: Direction argument:
$Z=3x+y$ grows with $x$ most quickly. So to minimise, take $x$ as small as allowed, namely $x=0$.
Step 2: On the y-axis:
With $x=0$ the constraints give $1\le y\le2$. $Z=y$, so the smallest is $y=1$.
Step 3: Check the other edge:
On $y=0$ the constraints give $1\le x\le3$, so $Z=3x\ge3>1$.
Step 4: Answer:
Minimum $=1$. Option (C).
Final Answer:
The minimum is at the corner (0, 1).
\[ \boxed{C} \]