OK, I had some tea, which is almost like caffeine.
The essence of what you want to do is install some kind of http authentication plugin. I took a quick look, and found this > http://dev.wp-plugins.org/browser/http-authentication/
I haven't tested it, and I don't know if there are other better ones.
Once you have that, an rss reader can put their user/pass in the url as I described above. (http://user:firstname.lastname@example.org/blog/feed). Use the actual url to the your feed, and just insert the user:pass@ bit after the //.
You could test this in a browser, except for IE, as I think microsoft disabled that function. If you're *logged out*, and try the url in firefox, you should get the xml for the feed.
Give that a try and report back here. I'll advise if you have a problem. Hopefully it will just work, so I don't have to actually try installing it myself! ;)