Hi there and
to the forums!
Luna is a very handsome little boy
He's around five months old, from what I can see. He's a recessive pied mutation budgie so his cere will be pink his whole life, and he won't develop the grey irises that adults have. He'll look like a cute baby forever!
Do you have toys in Luna's cage? Budgies need lots of toys to stay happy and active. Avoid toys made of rope or fabric, instead put toys with wood blocks, plastic rings, or shreddable yucca, etc. Also, you should remove all the dowel wood perches and replace them with natural wood perches. Dowels can hurt his feet and they eventually lead to bumble foot, a serious foot condition.
Here's a link with more information:
As for taming, I'm glad you haven't tried to forcibly touch him or take him out of the cage, etc. It sounds like he's used to your presence but doesn't trust you yet, which is normal for only having him one month. Every day, you should sit by his cage and read or talk to him for as long as you can until he's very comfortable with that. Continue this as often as you can as it helps to build trust, but after he is comfortable with you, etc. then you can additionally start putting your hand on the outside of the cage until he's okay with that, and gradually (over a period of weeks) put your hand in the cage far away from him first a little every day until he's used to it, then when he is, move slightly closer, and when he's used to that, closer, etc. Eventually after several weeks of this gradual routine (it could be months, every budgie is different), you can offer him some seed or millet from your palm and he can get used to eating from your hand. Even when he's comfortable with your hand, he may not want to come out of the cage. That's fine. The cage is is "safe place" and you shouldn't force him out, he'll come out when he's ready. After he's comfortable with you and your hand, you can leave the cage door open for one or two hours while you're in the room to supervise (and it's been budgie proofed, so no other pets or dangerous things) and he may decide to come out then.
I know that was probably a lot to read, haha. For further information check out the other budgie articles and "stickies" we have on the forums to ensure you're up to date on the very best of budgie care. If you have any questions after reading through everything, be sure to ask as we'd love to help! On the forums, we have all the information you need to know the best practices for caring for Luna.
We hope to see more of you and him around the forums!
Cheers and best wishes