Reshaharu

1. Go for the Ryzen 1600. It's so much better
2. Look into the 1060 6gb or 3gb GPUs as well
 
Overall looks pretty good

Put it together on pcpartpicker.com

You could swap the i5 for an R5 1600, get rid of hyper 212 and use a Gigabyte AB350 gaming 3 as the motherboard and it should be around the same price. You won't need a cooler other than the stock one if you don't plan on overclocking heavily. This route would be more future proof imo. 

you don't need an after market cooler as you cant overclock that cpu  

Ah, alright. Wasn't expecting those parts to be so much. The i5 should still be fine depending on what games she wants to play. 

I would say the 4GB 570 is better than the 3Gb 1060. Technically, the 1060 3Gb is a little faster, but I'd be wary of only 3Gb of Vram. Either card will do well however. 

The 1600 and the Gigabyte board come to roughly £300 in the UK, which converted to euro's is €340. The two prices you gave me is €353, so it's not too bad.