Your user stories should be rewritten from user's perspective, not technical team's perspective. See below.
US1: as an admin I want to write a message on a new page dedicated for this matter.
As an Admin, I want to be able to write a message in a full screen, so that I don't get distracted by other items.
US2: as an admin, I want to display my message on the big screen.
As an Admin, I want to be able to see my message on "big screen" so that I can ....
US3: as an admin I want to display my 5 last messages.
As an Admin, I want to be able to see my last few messages, so that I don't have to search through the whole list of messages.
US4: as an admin I want that all messages be saved so that the last 5 messages can be displayed as soon as the app is opened on the big screen.
As an Admin, I want to see my last few messages when I open the app, so that I can take a quick look at what was my last message.
As of now, your stories do not contain enough information for anyone new to see what is being built by the team. These stories lack transparency, because I need to understand what is "big screen", "why exactly 5 messages" etc. What you have written currently are the tasks that go in the user story.
Now, coming to the point of database building, it should not be first or last. Any technical artifact whether it is codebase, database, design documents, coding guidelines, functional tests etc are all living things. As and when your user stories are being worked on, database should be modified as required. Otherwise you will not be able to use the learnings in the future stories. This is also why user stories should be understandable to anyone vaguely familiar with the application, e.g. CEO, IT director etc.
So, the short and sweet answer to your question is, "don't focus on when to build database. let the team do that as and when needed". You should focus on just the functionality from user's perspective.
Another thing I noticed just now, user story 3 and 4 are actually the same thing. They can both be clubbed as below:
As an admin, I want to be able to preview last few messages so that I don't have to search through the whole list of messages.
Now, you can add "Must display preview on login" as one of the acceptance criteria.