SQL Server listen on a specific port
You can specify the port SQL Server listens on for each IP address it is listening on, but not per database.  To change an existing installation of SQL Server, run SQL Server Configuration Manager, expand SQL Server Network Configuration, and select Protocols for MSSQLSERVER in the left pane.  Then right-click on TCP/IP in the right pane and select Properties.  On the IP Addresses tab, change the TCP Port for each address as desired.

You will need to restart the service for the change to take effect.

Most SQL Server clients will use port 1433 by default, so if you use a different port, clients will have to be configured to designate the port SQL Server is listening on.

One of the issues that can occur with specifying an alternate port is that another application is already using that port, or RPC may grab that port before SQL Server starts up.  Google "rpc dynamic port range" to research that.  So if you configure SQL Server to use a port that is in the range that RPC uses, SQL Server may start one time, but not the next.  Use the netstat command to see what ports are in use on the server, and review documentation of installed applications to ensure you don't select a port that your applications are using.  Just because you don't see it in use in netstat doesn't mean the application doesn't use it itermittently--not likely, but possible.

Also, firewall rules on client, server, and network devices that allow port 1433 may need to be modified or added to so that the port of your choice will not be blocked.