>>1936435>Is a €500 budget viableYes absolutely.
Consider you will probably spend €100-200 to overhaul a used bike, with quality consumables. You want to pay the money for high-spec tires, new cables, new chain, new bartape etc. And service all the bearings. If it is an older bike you may well want to convert it to have wider gearing and better brakes.
Try get something for 200-300
That is the usual value of a decent high-ish end project, modern, or classic. You may well get a deal for less.
Do you want an 80s/90s classic like pic rel, or something more modern with shifters integrated into the brake levers? Both are good options. The old bike is a little more durable, charming and has cheaper parts, while the modern bike is lighter and has shifters on the bars.
Post any bikes you find for hot takes.