Oh dear!!!
When my budgie got sick while i was away, my mom was the one who had to take him to the vet for me and then administer the meds and extra fluid nutrition. She isn't exactly trained in that sort of thing (never was the one to clip the wings or nails) but she was just really careful and followed the instructions of the vet. She held him in a towel (tried to cover the eyes as much as possible), in a darkened room, and the slipped the tip of the syringe into his mouth, by going in from the side. You can also gently hold either the top or bottom of the beak to help get his mouth open enough. The first couple times were a miss because she wasn't quick enough and ended up spilling half of the fluid onto his chest. But she soon got the hang of it...so i'm sure you will too, especially since you have much more experience handling him.
Birds can die from stress, but remember that bud trusts you so the whole experience won't be as bad as if he was in a new place with a stranger handling him. Just try to do it as quickly and gently as possible and then return him to his warm cage to rest. Because he is sick, he shouldn't put up as big of a fight as usual.
Did the vet instruct you to add a warm water bottle wrapped in a towel to the cage?
I don't know how hot it gets during the day where you are, but its not unbearable, then I would suggest you do that as Bud will probably like to rest on top of it. It will help warm up his chest to fight off the infection. My budgie pretty much stayed on it 24/7 for the first few days, even though the house was warm, and even allowed us to cover him with a smaller towel. After the 5th day or so we finally heard a little chirp from him as he started to perk up!
Hope he makes a speedy recovery!