Can Far Cry 3 play on a AMD Radeon hd 7570?

Nope.

http://www.tomshardware.co.uk/far-cry-3-performance-ben...

Even at rock bottom settings you'd need at least a 6670.

In have a 7570 that I am using while my 7870 hawk is rmaing, but I do know that fc3 does run at about 30 fps. Similar to bf3 performance. If I were you, I would build the computer because a 7570 is not a card you will want to stay with for long.

Just remember anything x5xx or x5x and under is crap, always has been since the beginning modern gpu models, say 15 years ago. Generally x6x and x7x are the mid-range, and x8x and x9x are high

My old 9400GT runs FC3 at Lowest possible (800x600 on Optimal).
I get a very smooth gameplay, I don't feel that I'm playing the game on 10fps.  

I compared it to my GTX 660 (On Ultra 1080p) vs 9400GT (800x600 on Optimal) , I found that my 660 is alot smoother but its really hard to tell that which is better in the terms of gameplay until you see the fps.

?? I have HP Envy with Beats Audio + AMD Radeon HD 7570 and this play far cry 3 no problem... I have no idea why you would say it wouldn't... The game is so smooth and crisp..
what is your other option apart from the p rebuilt you mentioned? I payed 760$ taxes in for the desktop *no monitor* on boxing day and i have to say that this blew my mind for audio and game play for the price. i do have 1tb hd, 10gb of high end ram and cpu is sick but my next upgrade is getting another monitor then another graphics card.. Just shows you that i would rather have dual monitor then forking out the cash for a new card..
Don't listned to people who never tried it before, test it out yourself then make your choice. For every hatter out there you'll find an actual person who uses the card that say the complete opposite..

ps: I played black ops 2 at max and that is even more beautiful to look at... borderlands 2 also another one.