If a university is named by a place, like "the Univerisity of Bonn". Can I call it "Bonn University"? When saying the former, is the mistake serious if without definite article "the" before "university"?
No. The University of Chicago is a proper name and is not known as the Chicago University, though it may be a Chicago university among several. Rice and the University of Houston, for instance, are both Houston universities. Boston University is a proper name, however, and you would not refer to it as the University of Boston.
When a university's proper name takes the form X University/College, it's common to call it X:
They studied at Ohio State, Texas A&M, Harvard, Swarthmore.
If the official name of a university includes the name of a city, like Boston University, you'd likely use the full name to avoid confusion.
The proper name in German of the University of Bonn is:
Rheinische Friedrich-Wilhelms-Universität Bonn
The Rhine Friedrich Wilhelm University of Bonn
In older works in English, the name was also translated into the Frederick William University of Bonn. The name of the river was added because at one time there were three German universities named for Kaiser Friedrich Wilhelm III, one in Berlin, now the Humboldt University, and another in Silesia, now a part of Poland. This name rarely appears in the press, where the names of universities are often shortened to Uni Bonn, Uni Trier, etc.
This means you should still refer to it in English as the University of Bonn.