Working with the info you have given me, it looks like that PSU should handle that rig fine. A word of warning however; although that PSU isn't exactly one which would be likely to burn your house down, They aren't exactly the best out there.
As I said, it should be fine. However, if you invest in a new PSU in the future, try to use brands such as Corsair, XFX and antec. These use seasonic or other very good PSU OEMs so you will be getting a very reliable PSU. One of the most important rules of computer building is to not cheap out on your PSU! Just keep that in mind when you buy a new PSU. I have personally had bad experiences with BeQuiet
Let me know if you need any additional info in order to determine your answer, i don't undertand much about hardware i just wrote what i thought i should.
it will work since both the CPU and GPU are not power hungry but for how long depends on the 300w PSU quality. I assume it's from a mass produced PC (Dell, HP, etc) so I would start saving for a quality PSU sooner rather then later if I was you.