To find the price per cup of the non-dairy creamer, we first need to determine how many cups are in a 2-quart carton.
There are 4 cups in a quart, so in a 2-quart carton:
\[ 2 \text{ quarts} \times 4 \text{ cups/quart} = 8 \text{ cups} \]
Now, we take the total cost of the carton, which is $9.28, and divide it by the number of cups:
\[ \text{Price per cup} = \frac{\text{Total cost}}{\text{Number of cups}} = \frac{9.28}{8} \]
Calculating that gives:
\[ \text{Price per cup} = 1.16 \]
Therefore, the price per cup of the non-dairy creamer is $1.16.