The weblink component (com_weblinks) is a part of the Joomla! core and is distributed with the basic Joomla. As such it is not updated separately so there is no “updated version” available other than the one that is appropriate for the version of Joomla! you have installed.
Since you mentioned you used the “one-click” installer, you should be using Joomla! version 1.0.13, and the appropriate weblinks component should be installed automatically.
if you configured Joomla! to only display the “weblinks” component for registered and logged in users, then it looks like the template may be the issue. If not (you have properly configured the weblinks component to be displayed for “all” users (public), then you may have a deeper problem.
My first suggestion is to use the Joomla! Admin section to (temporarily, at least) set the default template to one of those that shipped with Joomla!, and see if that makes you able to properly load the weblinks component. If it does, you have a template problem. If it doesn’t, something else is likely amiss.
Make sure you “refresh” and/or clear your caches (depending upon how you have Joomla! configured) when testing with one of the “default” templates that was shipped with Joomla! so you get accurate results for your tests.