Mac OS X 10.8 App Store hogs the connection when downloading new OS X I'm trying to download Mavericks from App Store but it's hogging my internet connection.
Is there anyway to set the App Store to download at a lower speed or take fewer connections or whatever to be able at least to open web sites while downloading?
Not really - the OS is designed to do all the network transfer that the routers will allow. From a systems perspective - the network is so slow compared to storage, CPU, memory - that any attempt to hobble that subsystem is really a poor use of the computer.

You would need to have traffic shaping tools on the router or connect the Mac to the router using a slower interface if that would help.

The easy solution is to pause the download until you can allow the download to take what bandwidth it negotiates with your router and the Apple servers providing the download.