in the event of a smog alert, vehicles with license plates with even numbers at the end may only drive in Budapest on even-numbered dates and those ending with odd numbers on the other days.
Dust levels are quite high all over downtown, as you can see at this great website. Hosted by the environmental ministry, it shows continuously updated pollution levels at eight monitoring stations throughout the city. Incredibly, the site is in English. The page on the link shows a map of the city, with the monitoring stations indicated as blue dots. All you do is roll over a station and an information box pops up showing levels of several kinds of pollution. The pollutant you should look for is "PM10," which means particles less than 10 micrograms in diametre, or in plainer English, dust.
From what I read, motor traffic in urban areas is often responsible for two thirds or more of air pollution. It follows that a switch from car to bicycle would be a big help to air quality. Unfortunately, those who take a pioneering roll in the "velo-lution" have to put up with some pretty crappy air.