Did the vet take an x-ray or do blood work? It looks to be a neurological issue or an injury to his spine/head/neck. He needs an x-ray to rule-out a fracture, tumor, etc.
Budgies can live into their 20's, 10 years old is not at all "old" for a Budgie if it is fed a healthy diet; average lifespan in captivity if they are well taken care of is around 15 years or more, so this isn't a situation where you should just assume that the bird is "old" or "elderly" and that's what is going on. This is a common misconception, a lot of people buy Budgies thinking they will live to be only 5 or so years old unfortunately. My point is that though you don't want the bird to be in pain and suffering, you do want to make sure that an experienced Avian Specialist (hopefully), or any vet available, does the appropriate tests, the first of which should have been a regular x-ray. That way you can make an informed decision about what to do...If it's not a compressed-nerve, bulging disc, or any other condition caused by inflammation in his spine/neck, then the steroid isn't going to help, which is why they should have done an x-ray first...Hopefully they also gave him some pain medication, like Metacam, as he's obviously in tremendous discomfort...
Is the bird allowed out of his cage, or was he out of his cage prior to him acting this way? Did you hear any commotion inside of his cage prior to this? It really does look to be some type of acute injury....I hope that you can get more answers from either a different, more experienced vet, or can take him back to the same vet and request an x-ray before anything else to hopefully get a picture of what is going on, otherwise it's impossible to make an informed decision. If it's something due to inflammation or a pinched nerve, etc. that will resolved itself then you need to know that, versus it being a growth/mass/tumor that has been slow-growing and finally put pressure on his spine/neck/nerves and is not treatable.