How much time do you have today to spend with him? I ask, because if you have an hour or so (maybe more, an hour goes by fast), where you could take him out of the cage and set him out...say on your bed and just sit or lay there and let him walk around and explore. Lay a towel down, sprinkle some treats, even if it's just some of the food you are feeding him, and read, draw, color, if you have TV/radio in your room, you could even turn it on for some background noise if you want. Then just "chill" with him for as long as you can...oh and keep some wipes or toilet paper near by so you can wipe up any poo's.
He might not do a lot at first, he might just stand and look around, expect him to be nervous...this is all new to him, and to you and he senses your fears too, so try not to be scared. You will do just fine with him.
Talk to him a lot, hello, say his name, such a good boy, you're so pretty, awww...look at you! Things like that, he'll relax and start exploring and before you know it, he'll be climbing all over you.
Now, I know many people will disagree with me here....but it's worked for me for every bird we have...Chicken Little was completely wild and would take chunks of flesh if given the chance....so when I took him out of his cage, I used a little towel and grabbed him, (so I wouldn't get bit) then once he was out, he was GREAT! He LOVED his "walkabout" time. I always offered my finger first for him to come out and go back to his cage, if he resisted that, then I'd just towel him out and back into his cage if necessary.
Within a couple weeks he was happily hopping up on our fingers to come out and go back. We never towel now...no need to.
Again, I know many will disagree with it, but if you offer your hand for him to come out and he takes off to the back of the cage, or he just won't come out of his cage even if you leave the door open all day, and especially if he'll bite HARD if you try and hold him, then toweling is an option that I have found to be effective and has given us solid results to hand taming our birds.
So in answer to your question, how long before you take him out of his cage...I'd say, what are you waiting for? Take him out and spend as much time "chillin'" with him as you can. Sometimes, the transition of going in and out of their cage is a scary one but if he enjoys his time out with you, then it will only be a matter of days, maybe not even that long, before he's waiting at the door for his quality play time with you.
Have a great day with your new baby!
Toni