Alexei is tasting the sailors' lunch on the Standart. It was a habit of the Emperor, which Alexei undertook as well, to taste the lunch of the regular men serving the Emperor, every day. It was a way of showing the regular men that THEIR food was good enough even for the Emperor himself and a way for the Emperor to show his concern for his men, by showing he cared enough to taste their food to make sure it was good enough for them.
Nicholas, and then Alexei as well, partook of this ritual every single day, wherever they were.