I'm also giving my budgie, Sprite, some medicine at the moment. It was definitely a bit challenging at first, but it's not that hard once you get the hang of it. Also, your bird will learn what's happening after a while, and won't freak out as badly.
What I do is I get a small towel, and hold it in my left hand (non-dominant). Then I use my right hand to get my bird to step up, and I put her up to the right cage wall, and let her hold on to it. Then I just gently grab her from behind. I do need to be quick though, because she's very clever and she will get away if I'm too slow.
Once that's done, I make sure I am restraining her wings so she can't flap and get away. You need to hold firmly, because they are stronger than you think. But definitely still be gentle. You can also use your thumb and forefinger to keep his head in one place, but don't actually hold his head. The most important thing is that you don't squeeze the chest, because otherwise he won't be able to breathe.
As soon as you get the hold right, put the syringe up to his beak. Sprite bites the syringe, so it's not hard to get her mouth open. I don't know if your bird will do the same. What's important is that you put the syringe into the side of his beak. Never come from the front, because then there's a chance he could could breathe it in. The airway is under the toungue, so just be aware of that.
Your bird will struggle to get his head away from the syringe, so it may take some time before you can get the syringe in the proper position. Don't rush.
As soon as the full dose is in his mouth, put him back on one of his perches. I always give Sprite a big reward afterwards, which definitely helps to make the experience less stressful and scary.
. You just have to sneak the syringe into their mouth.
I found the best method was to cup her in my hand with her feet holding onto my pinkie finger and her head poking out between my thumb and index finger (that way she couldn't try to fly away). Then I would use my index finger (on the hand she is currently in) and put it near her mouth, since she was so angry that i'm holding her, she bites my finger. Instead of pulling away, I carefully keep my finger in her mouth (and my thumb is at the back of her head so she can't pull back. I'm not forcing her head back with my index finger though or squishing her head, I'm just not allowing her to pull her head back since my thumb is now in the way.. if that makes sense!) and am able to get the syringe in since her beak is now being held open. It was A LOT better than trying to use the syringe to coax her mouth open and went so much faster. Of course, there is a balance between being forceful and gentle and I tried to be somewhere in the middle. I found this went quicker and was therefore less stressful. Best of luck!!!!