The processor performance of an Athlon 64 x2 4000+ would likely be an issue in many newer games. One of my computers has a similar Athlon 64 processor, and they just don't have the per-core performance to handle most AAA games from the last few years at reasonable frame rates. Even low-end current generation processors offer around 2 to 3 times the performance per core.
It's difficult to say how well AC IV might run. An RX 460 should be plenty capable at handling the game fairly well at 1366x768, but the minimum requirements recommend a quad-core CPU with somewhat higher per-core performance, so the game's performance would definitely be CPU-limited. It might be playable, but I would expect the frame rate to often drop below 30fps. Also, being an older system, if the motherboard only has PCIe 1.1, it might not be compatible with an RX 460.
I agree with the recommendation to upgrade to a new CPU, motherboard and RAM. If you're on a tight budget, I would currently go with a Pentium G4560 system, like helpstar suggested, since it should be nearly as fast as an i3 6100. That will run you around $200 though, in addition to the graphics card.