If the content of the linked docs isn't displayed in the doc you're printing
from Word isn't going to "just print it" - all Word knows about those target
docs is where they're supposed to be, what they're called & that it has
little buttons (icons) that will reach out to open them
If the linked
docs are displayed as icons the "Update..." command doesn't really do much
of anything. OTOH, If the content of the linked doc is displayed, what
prints will be the most up-to-date information even if you haven't otherwise
updated the links prior to printing.
In order to convert the icons to actual objects you'd need to do so manually
or programmatically before printing. That could present a number of
additional challenges... especially if any of the linked files are more than
one page.
If you provide more detail on what you're working with & what you hope to
accomplish I'm certain there will be some suggestions offered that will
prove to be more satisfactory.