Why did Alex go to Daley Plaza the first time

It's hard to explain in written form but I'll try...there was no "first time". The "first time" we saw it was 2006 real time, BEFORE Kate "met" Alex, but two years after Alex met Kate. The whole movie preceded that event. Kate's timeline in the movie was VDay 2006 to VDay 2008. Alex's was VDay 2004 to VDay 2006. So when he dies in an early scene, everything you see after already happened to him, including the letter he remembered while talking to his brother, the letter telling him exactly where she was on VDay 2006. He goes there to see her, to meet her after "standing her up" when they were supposed to meet at the restaurant. The movie is fiction (obv), but the notion that a person in the future changing the past is not an unusual one in movies. On VDay 2008, she was told he was killed and went to the mailbox to warn him. Her letter says "I know why you went there, you went looking for me, and if you love me, don't come, wait for me." Bc he went home to find the letter (in 2006) he got her letter in time....apparently he still went but stayed back and didn't try to cross to see her.