Is a 350W enough for this?

Not that 350W PSU, and even with a good one it would be cutting it close. I'd recommend at least a 400W PSU, and preferably 450W or more.
 
The PSU that comes in that case is a POS, it wouldn't be good for much anything. I'll echo Zero and say look at getting a 400 or so watt psu.
 
If that included 350W PSU was of good quality, I'd have said yes. But since it isn't, I have to agree with Zero82z's assessment: Get a good quality 400W PSU. I recommend this cheap but good quality PSU:
$50 - Antec Earthwatts 430W PSU

650W is definitely overkill though (just in case that wasn't sarcasm from s1ngular)
 
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