You can get by with a 7750 with that psu.
If you want a better card and have a budget for psu, then get one, but not a cheap no name brand psu, but a decent one with 450-500W, like brands from corsair,xfx,seasonic,fsp,ocz,antec.
In your opinion is it worth getting a cheap 500W PS or something like that to get a more demanding video card, or could I 'get-by' with a video card running on 300W?
depends on your definition of "cheap"; a $20-40 500W PSU from a crappy brand is something of a risky purchase; at best, it won't produce sufficient power, at worst it destroys your whole PC. you're looking to spend $50-60 at at most for a decent 500w PSU, more if no good deals are abound.
as djangoringo said, the 7750 is about the only good card for a 300w PSU and it performs exceptionally well for the job. trying to go with anything beyond that with your 300w psu is something of a bet, as PSU's from a pre-built tend to be really crappy.