The wattages on power supplies are ususally some fantasy figure made up by someone that wouldnt even know their amps from their arse.
You should be fine on that minimally specced out PC with a real 300 watt power supply - Hell, I have more crap on my desktop PC that only has a 225w psu in it, but it can actually deliver it.
I would guess that your power supply came with a case, and is worth exactly what you paid for it - nothing. Any actual power supply should be able to run a machine like that. If you had a 3d card, and multiple harddrives then you start to need to think about going 400+ watts, other then that its a small draw.