By default, Arena passes priority during upkeep & draw steps unless there's a trigger. If you want to cast Orim's Chant on your opponent's turn before they can cast anything you need to have placed a stop on their upkeep before you end your turn.

in other words, it doesn't allow me to cast during their upkeep, the first time it allows me to cast it is as the other person is casting their first spell in the turn

The blue outline light doesn't come on during upkeep :)

You have to have priority to cast Orim's Chant. And during any player's turn, that player gets priority first during any step and phase where players get priority. So if you want to stop your opponent from casting a sorcery timing spell on his turn, you have to cast Orim's Chant during the upkeep or draw step of his turn. If you wait until the main phase, he has priority, the stack is empty, and it is his main phase, so he can cast a sorcery timing spell before you can cast Orim's Chant.