Dear @Florianus,
Welcome to the Wefugees Community and thank you for your question.
Yes, in general it should be possible to change your residence permit according to 16b (student visa) to one according to 16a (apprenticeship/vocational training). Did you actually see my colleague's answer in this thread, in which Christine is giving important information on the necessary steps? She also highlights that you should be careful not putting your current residence permit at risk in the meantime.
I think it is the best to contact the immigration office (Ausländerbehörde) directly in order to understand the exact documents they will require from you - but for an orientation you may see Berlin's immigration office's website and their list of requirements.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,
Meike