I've docked my ship at the landing pad of my outpost and proceeded to the crafting station inside the base, where I also have storage containers filled with materials. When I begin crafting, will the materials in the storage containers and ship be automatically accessible, or do I need to transfer them to my personal inventory first?
You cannot access the resources in storage containers located at your outposts from anywhere else. You can only access it by travelling to the storage containers yourself and "open" and "transfer" the content.
However, what you can do is build Cargo Links to transfer the resources automatically to a central place. By doing this, you only have to check the central place for newly arrived resources so you don't have to travel around to every single outpost.
I recommend to read some Outpost building and Cargo Link How-to's. They describe the needed parts in details.