Ocean City is a DRY island. That means you won't find any bars or liquor stores on the island. You'll easily be able to find a liquor store and bars right before you get on the bridge to head into OC. Most restaurants are BYOB, but I would suggest either calling or checking online before going to get your morning mimosa! :)
Ocean City and Margate would my suggestions! not too far, has walkable things in area and also cute!