I have a script that prompts for file selection, then imports and processes the contents.

One of the fields is a date field. Not today's date - as I don't always run it every day, but there's a file per day. The date is however contained in the filename, so I want to be get the name of imported file.

If you use Insert File and put the file in a container field as a reference only, you will have the file name available to you to parse.

Can I then Import data from the Container field?

Not quite. You now have the path to the Import File. If you set it to a variable, $import_path, then you can use in the Import Script Step's file path dialog.