To find the price per cup, we first need to determine how many cups are in a 2-quart carton.
1 quart is equal to 4 cups, so:
\[ 2 \text{ quarts} = 2 \times 4 \text{ cups} = 8 \text{ cups} \]
Next, we take the total cost of the yogurt, which is $4.56, and divide it by the number of cups:
\[ \text{Price per cup} = \frac{\text{Total cost}}{\text{Number of cups}} = \frac{4.56}{8} \]
Now, we can perform the division:
\[ \frac{4.56}{8} = 0.57 \]
Thus, the price per cup is:
\[ \boxed{0.57} \]