Hübner belonged to the SC Karl-Marx-Stadt sports club until the early 1990s and was trained by Siegfried Möckel (born June 12, 1950) for a long time. He also looked after him as the GDR association trainer and finally led him to his first world title in the sprint in 1986 .
On the other hand, Hübner was denied participation in the Olympic Games during his entire career as an athlete, as he had an overwhelming opponent in his own country in the two-time Olympic and four-time world champion Lutz Hesslich . Huebner lost to Cottbus in both 1984 and 1988 at the GDR championships, which were decisive for the award of the only starting place at the subsequent Olympic Games. Although he traveled to the Olympic Games in Moscow, he was a substitute for Lutz Heßlich there. In 1986 he was awarded the Patriotic Order of Merit in gold.
After the fall of the Wall in 1990, Huebner became a professional and immediately won the sprint world championships as such. In the same year, he also won the first of three consecutive world championship titles in the Keirin . In 1995 he was also world champion in the team sprint .
His many successes also include the four victories in the now-defunct classic Grand Prix de Paris , in which - contrary to the separation of amateurs and professionals that was practiced until the early 1990s - the starting field had consisted of drivers from both camps since 1975. Hübner won twice as an amateur and as a professional, making him the second best German cyclist. Lutz Heßlich won this competition six times. In addition, Huebner won a large number of international Grand Prix competitions for sprinters. So u. a. the Grand Prix of Copenhagen (three times), Hanover (three times), Germany (twice) or the Champions of Champions Trophy at the London Velodrome in Herne Hill (twice). He won other international sprint tournaments in Italy, Hungary, Austria, France, the Czech Republic ( Grand Prix Framar 1984), Germany, Poland and Denmark.