Can I use this with an iPaq h4150 to access my PCs Internet connection while in another room?

Yeah. You'll need to run some sort of gateway software or ICS on your PC though. You'll just need to run your wireless LAN in adhoc mode.

So I can use this even though I have no other wireless hardware (like access points on the router)? I don't have the iPaq yet, so would it be quite simple to set up such a gateway?

At the moment the cable modem is plugged into a Netgear RP114 router through which I feed 3 PCs.

Yeah. You shouldn't need anything else. If you've got a recentish version of Windows then you can just use Internet Connection Sharing (ICS) You'll need to have this computer on all the time you want to use the iPaq wireless though. If you don't want to have it on then you'll have to buy an access point to replace it.

Most wireless equipment allows you to operate in two modes. Infrastructure which requires access points and adhoc which does a sort of peer to peer arrangement with the computers.