Harry starts out as we all tend to, resisting changes, newness, things to come. This film is also a "coming of age" film, where the character finds peace within himself and his place in a world that stops for nobody and nothing. Note that early in the film the cops have to carry him bodily out of his apartment... that's how invested Harry is in things remaining the same.
By the end of the film, after seeing the old fade away and the new ever evolving, Harry has gained acceptance. "Goodbye, old friend" is as much a "thanks for the good times, pal" as you're likely to see in film. Overt mourning would have cheapened the scene and driven the film off its path. "I'm into Zen these days" is Harry's way of telling us it's all right. For him and for us. That we can accept what happens, what opportunities arise, what vicissitudes beset us.
He shows us with Tonto's passing as well as tells us with the cat lady.