Aww! Blue is just precious! From what you have written I don't think there is one thing you have missed to get him started living his life with you. It is very normal that he isn't going to eat and drink. I've had budgies that have gone for up to four days without making a sound or eating and drinking, so I know your concern. You could get some millet sprigs and hang one of them in his cage close to the food cup, and also, since he is out flying and is setting on your finger, gently offer him that millet sprig in one hand while having him perched on your other hand (finger). He just might take a little of it. Millet is good nutrition and also has moisture in it, so if he hasn't had any water, the moisture in the millet will keep him hydrated and the seed will give him a little nutrition.
I feed a mixed seed to my budgie and cockatiel. My birds also love red and green leaf lettuce, cilantro, dry cereals like cherrios and grapenuts (not sure you would have them in your country), but dry oatmeal, or quiona, or even wheat bran or wheat germ would be fine. You can mix some of those in the seed cup, or give him a separate seed cup with those dry cereals to him. I also feed pieces of whole wheat bread as well. I have found budgies are not into fruit much, so the fruit seed stick you got him was a great idea. Raw apple pieces are a hit for my birds. I wash the apple thoroughly and just cut it up in small pieces and place a piece between the cage bars. You also could feed the apple peels if you wash them thoroughly as well. Never feed apple seeds or other fruit seeds to him, because in my research, I have found they can be fatal to birds if they eat them.
You might consider googling "good foods to feed budgies". There should be some great information to read. And while you are googling good foods, be sure to google "foods" that budgies should not and/or never eat. (Thankfully we have the world wide web that has so much information on it for us).
I am giving you an A++ for all that you thought up to do for him. Sitting by his cage, the music, singing to him, talking to him, etc. etc. are super. My birds love to go on Youtube and see the birdies. We watch cockatiels and budgies, and they LOVE it! You can use the millet I spoke about as a training tool as well. But for now, I would just offer it to him so his little tummy will have a little food in it until he starts eating. Please let us know how things go for him. I fully expect him to be chittering and chattering to you in a day or so. Letting him fly free will also help him to trust your home and you and make him happy. Congratulations on bringing him home Blue! He is a beautiful and his name fits him perfectly!!
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