Is it just that the free Wi-Fi is particularly bad where you are ?
When Firefox doesn't get any web sites can you access the Ubuntu
Software Center or get updates ? If you can't do that either then the
problem is more likely to be the Wi-Fi itself rather than Ubuntu or
the adapter.

Any chance you can take this system to a friend who has a wireless
router connected to DSL and see if your adapter / Firefox etc works as
it should there ?