Welcome to the Wefugees community and thank you for your question.
Marrying your boyfriend won't get you a residence permit based on asylum. You'd have to go through an asylum procedure to get that.
And I also doubt that marriage alone is going to help you to stay in Austria once your visa expires because he is not an Austrian citizen. However, if his asylum procedure ends with a positive result and he receives a residence permit, you might be able to live together through family reunification (Or did he already get his residence permit? It's not quite clear in the initial question).
You will find extensive information about the family reunification procedure following this link: