The upgrade rig with Ryzen you suggested would work (note that Ryzen 5 1600 has been replaced with 2600 and 2600X, so you might look into them instead). You don't necessarily have to buy a new GPU now - you can reuse your old one until you save some money for a better GPU later. It would still be a very nice platform upgrade and you will most certainly see a performance boost, even with the same GPU. That would keep the cost below 500 EUR for sure. You can also reuse your old disks, and especially SSDs (if you have any).
By the way, which GPU exactly do you have? R9 200 series consists of many models. For example. R9 240 and R9 280X are not even comparable, so it would help to know better.