it works with a litte fix to play and host on one and the same mashine.
but don't try to run a server on your home internet connection because it's zu slow to host a counter strike server.
you will get a ping of ~500 with 4 players on it.
if you want to run a dedicated server go and rent a root server with highspeed connection (at least 100Mbit/s, but it should be 1GBit if you want to play with more than 10 players).
best thing for a CS1.6 Dedicated is a Linux Debian Sarge server, because you will get high performence on low prices.
if you don't know how to control a Linux server via SSH Telnet, then try to rent a Windows server.
but windows servers are easily hackable and cost the dubble of a linux server.
in germany a linux server costs 40€ per month which is ca. 35$ + 50$ installing for the first month
and a windows server cost 100€ which is about 90$ per month + 50$ installing for the first month.
on such a server with 1GBit/s connection you can run about 3 public servers each with 16 slots.
if you don't have any experience with rootservers, then try a server from gamedeamons.net
the main servers ware in Dallas TX but you can also have a server in New York, Chicargo, Atlanta or Seatle
i've never tested these server cause i'm not from the US but i heard many good things about them.