If it had good links and good content both.
If truly have good content and/or good links then you can outrank your competitors overnight. Otherwise, it may take months or even years to achieve that.
I ranked a small blog once. It took me one week.
It depends on the optimization techniques you follow. It can take a month or two for ranking your blog in search engines.
Usually for a small blog if the content post are engaging in the search results it will be provided more often to satisfy the users.
According to multiple sources, the average time for websites to rank on Google through optimization techniques is about three to six months.