I'm looking for and adjective which would describe someone who cares about something deeply, say grammar, and therefore is irritated by every little mistake when you write some text together. Something like "excessively prudent".
Pedantic is precisely the word for this. Webster adds the following note:
Pedantic is an insulting word used to describe someone who annoys others by correcting small errors, caring too much about minor details, or emphasizing their own expertise especially in some narrow or boring subject matter.