There is a supermarket on via Capo, a short way past, and on the same side as, Hotel Ascot. From Marina Grande, walk or bus up to the main road then turn in the direction away from the town centre.
There is also a reasonable mini-market nearly opposite where the road from Marina Grande reaches the top.
I have not stayed at the Admiral but I think it is a popular hotel. I'm sure you will enjoy your stay.
regards
Peter.