“went off” is right here, but “when it was set for the morning” is not quite right. To set, or to be set, is the act of programming it. You set the alarm at night, the alarm goes off in the morning. Also, you can just say “alarm” and most people would understand it to be on your phone (at least in the US, few people use actual alarm clocks anymore, although they do exist). I think you can just say, “The alarm went off in the morning” or “The alarm on my phone went off in the morning.” Obviously it goes off at the time you set it for, so you don’t need the second part.