Upgrading a GTS 240 to a Radeon HD 7770

Hello, I am pretty new to upgrading my computer hardware and am looking for some help finding a new gaming graphics card. I play a lot of league of legends, but am trying to find a card that will let my sister play the sims 3(something that doesn't seem to be working with our current card) and Far Cry 3. Not looking top of the line, but decent for about 140 bucks. I have an old dell from a few years ago. it has:

intel core i7 860 cpu
8 gb ram(4x2GB sticks)
nVidea GeForce GTS 240
a dell power supply labeled at 350 watts
my monitor is 1920x1080 if that matters at all

I am a little worried that the 7770 would require a little more power than my system can give, but I am unsure. Any incites or recommendations for other cards would be much appreciated.