You'll only need a quality 400w with that card. I have a 4870 (which is MUCH more of a power hog than that 5770) and it runs fine on a 500w with quad core overclocked proc.
Newegg.com - Thermaltake TR2 W0379RU 500W ATX 12V v2.2 Power Supply
This would be more than enough to power his system.
I think that MSI radeon is an excellent choice.
I would recommend upgrading his processor to a Phenom II x4 920 if you can afford it, but if you can't then his current Athlon II x2 215 with the 5770 should be enough.
I run Battlefield Bad Company 2 @ 1600x1200 with everything on max and 4xaa / 16xaf in Win7 with good framerate. I'm not sure exactly what my frames are but it doesn't ever dip far enough for me to notice it. Even when I play online and lots of things are going on, it's pretty smooth. The 5770 is a slightly slower 4870 with better overclocking headroom, draws less power, and supports DX11.
Oh and btw this is on a Phenom II x2 550 dual core. I have unlocked the cores, but before I did BFBC2 ran fine maxed out @ 1600x1200