He will get better with time. Birds need routine, everyday. That way they know what is coming and when. It makes them less nervous. So maybe start a routine in the morning, first thing. Even if it is opening up the cage cover and clucking to him. Or serving him breakfast when he sees you eat...
Millet sprays, pasta, lettuce, carrot shavings, seeds of any kind...my birds eat nearly EVERYTHING. If they see you eating it, sometimes this will entice them, or at least pretend to eat it.
Your budgie will catch on. Include him in on everything that goes on around your house and soon he...
I do think you may need to force him out of the cage, as he will continue to see the cage as his sanctuary and may never come around. So get him out and see if you can work with him out of his comfort zone.
But you are on the right track now. Remember, all birds have their personalities. I have...
I think you are rushing this bird. It takes time for a bird to climatize to a household, you and the things going on around his area. So I think you need to go slow with him. Get him used to you by teaching him the "step up" command. Get him out of his cage and put him on a play stand. Let him...