EDIT: Thank you to Unread Biscuit26
There is a rewind option on the main screen of Longshot once you have completed the game your first time. So you can reload the throwing competition with Dan Marino once you've beaten the game and get your missed achievement :) The bottom of the solution still applies if you want to keep trying to beat Marino on your first attempt!
I can confirm that if you quit before the timer is up in this little mini game and you reload your save, it will load you back into the cut scene right before the throwing competition starts. This way you can try as many times as you like without having to play through the whole story mode again like I did. Happy Hunting! :)