I very much doubt High-Ultra at playable frame rates. My own 965BE with a 7850 probably couldn't swing what you want this thing to do. Also, mine is overclocked slightly and a little more powerful of a video card, but when measured using a Watt's Up Pro watt meter it's pulling about 350-360 watts from the wall, which would mean that 350watt supply you have in there is cutting it really close to it's max capacity, I would seriously recommend you up the PSU capacity in that config.
My PSU is actually 85% efficient rated, which means 350-360 from the wall is 297 - 306 used. The Antec Basiq 350 only has 10A and 13A 12v rails, so only 276 watts on 12v lines and only if it's perfectly distributed. You also generally don't want to run PSUs over 80% of their capacity for longevity, so that 276 watts would be a more reliable 220.
Running that machine on it is pushing it really hard.
Even the Antec Earthwatts 380w has a 17A and 15A 12v set, which would make far more sense.
I have a 7770 and it doesn't quite reach ultra although it does just about reach high on 30 FPS on a 1920x1080 display. Anyway, I play on medium at about 40-45 FPS which feels smooth and runs well. However, the Athlon II isn't quite as strong as my CPU and BF4 is a very CPU reliant game.
Another point to mention is that if using 4GB's of memory when playing on ultra, it probably won't run very smooth.
This is false. 4gb RAM is plenty to run game at ultra. I recently had a stick of ram go bad and I ran at 4gb and maxed all my games still. Everyone that says you need 16gb RAM is completely wrong. Most games barely used 3gb.