The thinking behind that change in the law in Hungary is that, reports Veronika Gulyas in The Wall Street Journal, that drunk cycling, even on the main roads, won't hurt others. The assumption is that you only hurt yourself if you have a mishap when riding drunk.
Until this modification, since 2012, drunk cyclists had to stick to back roads. Here is the WSJ coverage (sub. req.)
Of course, not everyone, including in Hungary, is pleased with this decision. A drunken cyclist who gets into an accident could impede motor vehicle traffic in ways which trigger other accidents. Also, drunk cycling can distract motor vehicle drivers.
Other Eastern European nations such as Poland and the Czech Republic, have heavy penalties for riding bicycles with a certain blood level of alcohol.