Counter Strike:Source with it's greatly reduced DX7 path might play better than CS1.6 as 1.6 uses OpenGL.
Your main problem is your PC only has an SIS Integrated videocard, which not only blows for gaming but has very, very poor OpenGL support. It has mediocre at best DirectX support and generally does a tad better in DirectX vs. OpenGL.
That system does have an AGP Slot, so you can surely add a "real" 3d card in AGP form. Unfortunately, depending on what you consider upgrading to, you may also need to add additional case cooling (beefier 3d cards generate more heat) as well as possibly replace the power supply (many Compaq's have very weak and under rated power supplies- not enough juice to drive a real 3d accelerator).
I'd start by inspecting your power supply for wattage/amperage on the 12v rail (usually a sticker on the power supply inside the case) and what kind of budget you may have. From there, we can recommend you a plethora of AGP videocard solutions that should enjoy huge improvements in both CS1.6 and Source engine games.. as well as all other games.