Best case scenario you still have network trips going out to the service, so it's still not a great solution UNLESS the caching was done by your webapp - which is what I spoke about at the end of my comment above.
Unless this works w/o adding network roundtrips on each request, it's not a great idea.
I think the Unix philosophy of "do one thing and do it well" applies here. There are already off-the-shelf caching solutions that do what you describe: for example, with Varnish you can serve cached pages immediately and update the cache contents in the background.
It would probably be better to use those than reimplement them in an uber-webapp.
Unless this works w/o adding network roundtrips on each request, it's not a great idea.