Getting a faster wireless router (like the new Airport Extreme) would increase the speed of your wireless network (useful if you have another 802.11n device around the house you copy files to, like the

TV), but it shouldn't increase the speed of your Internet connection.
Your cable/dsl modem is the bottleneck there.
Exactly. That's why I proposed he let us know exactly what plan he is on. The test results are not really helpful since they are not accurate (at least not in my case).
But, generally, I have found that the local network even at 10-BaseT speeds, is generally faster than the fastest Internet connection. I am getting ready to upgrade to Cable with a 3 MB transfer rate, and that is still going to be well under the speed that an old 10BaseT connection would support.
For comparison, if my memory is right, 802.11b should be comparable to a 10BaseT connection.
802.11G should be roughly half of 100BaseT. And, I'm not currently aware of the maximum speed of 802.11n (haven't been following it that closely).
But, generally, if you have 802.11b or 802.11g, then you are already going to be fine unless you are one of the few who can afford an Internet connection that is actually faster than that.
Based on the speed you cite of 9355 Kb/sec from the test, that is still only about 1 MB per second. If 802.11n transfers roughly 10 MB or 11 MB per second, then that would still be much faster than your current Internet connection.
So, in short, with the little information you provide, I don't see the new Airport hub providing you with better performance.
Now, if you have a really, really, really fast connection that isn't reflected in your test results, then that might be different.