Definitely not advisable. There are plenty of reputable cab companies in Budapest and all just a quick phone call away. Hailing a cab on the street puts you at risk to unscrupulous and often unlicensed cab operators who will be more than happy to charge more than 10 times the normal fare using tampered meters.
Our favorite company, though there are many good ones, is (0036)(12)666666 (each cab has a dice on the top!), they can be relied on to turn up on early morning pickups, the drivers are usually courteous (though don’t count on it) and turn up quickly when needed….

Hungary operates of strict zero tolerance to drink driving. Whereas in most places you can be breathalysed to see how much alcohol you have drunk, in Hungary they breathalyse you to see whether you have recently drunk anything at all….