Does the hotel have a little shop inside the lobby where I can get a milk?

There is a Bartells Drugstore right next to the main entrance. They have almost anything.

If Starbucks is closed, there's a small shop 1 block away. Clays Market

In the Starbucks they sell packed beverages

Nope. But there's one closer to the hotel. There's Starbucks in the lobby.