Studying Language for International Students

I want to come to Germany early to finish off my language skills and get better in tuned with the German atmosphere before I start university. My plans are to register to a language Institute (already registered at the Anglo-German Institut in Stuttgart for April, just havent paid yet) and then apply for my studies whilst there. Is this a good idea or shuold I stay here and apply for University from home?

I really think that studying German will better my life in Germany for when I finally start University but another risk I am taking is what if I get there and none of the universities accept me before my visa expires and I cant apply for an extension.

I just need your personal  opinions and that would help me make my final decision.
Thank You

Application deadlines for university entry are far more than a month before a course starts. If you only apply in April, you may or may not be able to start school in November (start of winter semester).
Ask at the university of your choice about application requirements, procedures and timelines.
But in general: Yes, it is a very good idea to arrive beforehand to brush up your German - during the semester you might not have enough time for it!