I checked the users resource centre and there are thousands of hits on various combinations and iterations of network, directory, and cache/caching. Nothing jumped out at me. Sadly, I have no way of testing this for you as I don't have access to any network other than the one I run at home.
The concept is reasonable and you're right, it should be possible! But has it been "realised"? I have MANY file managers installed on my other system. Sometime this evening or tomorrow I'll try to read through the documentation for them and see if I strike gold. In the meantime, you could look at Altap Salamander (can use TC's plugi-ns), Total Commander (the plug-in system may yield something), Frigate 3 (has a fully featured web browser built-in), V the File Viewer, FileBoss (might be a strong candidate), FileQuest (though development has been stagnant for over a year), and DiskJockey File Viewer (also with built-in web browsing but also stalled development-wise for over a year). You could also check out AccelMan, which is now freeware. It, too, features web browsing (I think) and has a bunch of interesting features, if it is a bit quirky and in need of updating.
Sorry for the stream of consciousness response - good luck!
UPDATE: just checked and Altap Salamander has an option (enabled by default) Cache visited remote directories