You don't need to. Alan's advice is good though especially if you are in a more humid climate. I for instance don't bother plus I keep mine in the bathroom and don't use each razor I have for more than 1 shave a week if that. you can apply it by just dripping a couple of drops on the blade and smearing it around with your finger, but a cloth or tissue would be safer.
Part of the reason you are a little confused is that Lynn, who honed your razor, coats it in mineral oil for shipping. He wants to prevent rust, even if the mail man chucks the package in a puddle so he coats it. This should be wiped off prior to use.
As far as the case it came in, what kind of case is it? I never liked storing a razor in the cardboard box figuring the box could trap moisture, but some guys think the box pulls moisture off a razor so who knows :shrug: if its a wooden presentation box I would use it for storage most definitely. A plastic box will trap moisture unless its vented so that isn't a good storage option.