To get to Tokaj from Budapest, take motorway M3, then M30 to Miskolc (168 km), then road 37 to Tokaj (50 km). This road will get you to Tokaj from the south, on Bajcsy-Zsilinszky street. Click here for map.
You will need to buy an e-Vignette (matrica) to travel on highways. You can do this at the first stop across the border or at most petrol stations in Austria close to the Hungarian border. You have to buy an e-Vignette either for the whole country, or at least for three counties: Pest, Heves and Borsod-Abaúj-Zemplén, if you travel from Budapest to Tokaj.