According to the article, they have already demonstrated how to get a shell prompt on an unpatched OpenBSD box. Sooner or later there will be Linux, Solaris and other variants floating around the net, waiting to be used by the script kiddies. I've patched all my boxes last night (RedHat and Mandrake). Love urpmi from Mandrake :)
I think you are going to have to get shell access to upgrade Apache. The 'Hows' of the process will depend on how you have it set up now. If you have an RPM based server you probably would want to go that way again so all your webmin hooks keep working.
You would upload all the new apache rpms into a folder and then just go: rpm -Uhv *.rpm
If this is all new to you, you better get some help though.