Yes, Beethoven was alive during Mozart's lifetime. Beethoven was 20 when Mozart died. Beethoven had visited Mozart in Vienna and played him some of his compositions. Mozart recognised that he had talent and decided that he would like to teach Beethoven. Before their lessons could begin, Beethoven's mother fell ill and he had to return to Bonn to tend to her. Before he could return to Vienna to have his lessons, Mozart had died.
Further information
In 1787 (at the age of sixteen), Beethoven visited Vienna where he met Mozart for the first time. Here, Mozart actually gave Beethoven his first lesson in composition before Beethoven was called back to Bonn with the news of his dying mother. Beethoven's admiration of Mozart, and this first lesson in composition, is why music from the "first period" of Beethoven's composing reflects Mozart's style.