old monitor new gpu

DVI-D to VGA active
or
HDMI to VGA active

There are a lot. DVA, DVI, HDMI, etc. I assume your GPU has a DVI connector, in which case what you'd need is a DVI to HDMI adapter like this one. But there are a few things to consider before purchasing - does your GPU have a DVI connector? Or is it another one?

Also, some monitors may have a plug for DVI cables, so check first, it could save you some time and trouble if you just need the cable and no adapter.

It should, yes. That bulky box is the actual converter that transfers the digital signal to analog. The price is in range of what is expected as well. And AFAIK, R9 480 has at least one HDMI port. I'm not even sure if it has DVI ports any more, it is kind of an old technology now (albeit funny enough on most cards, the DVI port supports better bandwidth than the HDMI).

Edit: Just be careful: this adapter has to go into the GPU, so the bulky end and the short cable mean it needs to rest on something, otherwise its own weight may pull it out from the PC or lead to connection issues. Usually these short-cable ones are used for laptops or mini-PCs, where this cable can just sit on a table.