Ok good, I hoped you already had a PC to reuse.
Upgrading to any modern platform will require new memory as well.
I would not really bother with X serries for AMD, does not provide enough benefit for its costs, after all they are all overclockable.
If you dont need a new GPU then $300 should be enough but if you do need a new video card then that is still going to be tough to do on $300.
The Ryzen 1500 is better in multi-core benchmarks but another option to reduce costs would be a pentium g4560 build with b250 board and 8gb ram. That will run you $200 USD (not sure about local prices/currency). Then you could get a $150 GPU to go with it. This setup with say a gtx 1050 ti will be good for any game on medium to high settings at 1080p.
The Ryzen + a GTX 1060 or RX 580 will be good for ultra at 1080p.