There is a DADA syllabus, but Hogwarts teachers don't follow it.
Barty Crouch Jr. mentioned that the Ministry didn't think fourth years should be shown Unforgivable Curses yet, so that shows that the Ministry did have guidelines for what was supposed to be taught at Hogwarts, even before Umbridge came along and started making up new laws like she was being paid per new law created.
However, the answer to "Does Hogwarts follow the syllabus?" is undoubtedly no. What students were taught in Defense Against the Dark Arts varied widely, from Lockhart's class on himself to Barty Crouch Jr. teaching more advanced things than he was supposed to. From this, it's easy to see no syllabus was followed or enforced - despite its existence.
Instead, it seems like the teacher gets to choose what they teach.
Almost nothing Lockhart teaches is actually about Defense Against the Dark Arts. Instead, he turned it into a class on him. This seems like a very good indication that there isn't any set curriculum that the teachers were supposed to follow, because Lockhart was able to keep teaching his class on himself and didn't seem to be reprimanded on it, or start making any effort to teach actual Defense Against the Dark Arts content. Lockhart makes everyone study Lockhart's books, as the textbooks for at least three different years, probably for all of them. He also gives a test where the questions are all about him. This almost certainly wouldn't have been on any approved syllabus.
“Nothing to worry about – just to check how well you’ve read them, how much you’ve taken in …’
When he had handed out the test papers he returned to the front of the class and said, ‘You have thirty minutes. Start – now!’
Harry looked down at his paper and read:
1. What is Gilderoy Lockhart’s favourite colour?
2. What is Gilderoy Lockhart’s secret ambition?
3. What, in your opinion, is Gilderoy Lockhart’s greatest achievement to date?”
- Harry Potter and the Chamber of Secrets, Chapter 6 (Gilderoy Lockhart)
This seems to be a pretty good indication that the Defense Against the Dark Arts teachers are allowed to basically teach whatever they want. Those questions aren't even related to Defense Against the Dark Arts, not even tangentially.
Lockhart also reads passages from his books, and makes the students reenact the stories about him from in his books during their classes.
“Since the disastrous episode of the pixies, Professor Lockhart had not brought live creatures to class. Instead, he read passages from his books to them, and sometimes re-enacted some of the more dramatic bits. He usually picked Harry to help him with these reconstructions; so far, Harry had been forced to play a simple Transylvanian villager whom Lockhart had cured of a Babbling Curse, a yeti with a head-cold, and a vampire who had been unable to eat anything except lettuce since Lockhart had dealt with him.”
- Harry Potter and the Chamber of Secrets, Chapter 10 (The Rogue Bludger)
He makes Harry pretend to be a werewolf and they reenact the defeat of the Wagga Wagga Werewolf.
“Nice loud howl, Harry – exactly – and then, if you’ll believe it, I pounced – like this – slammed him to the floor – thus – with one hand, I managed to hold him down – with my other, I put my wand to his throat – I then screwed up my remaining strength and performed the immensely complex Homorphus Charm – he let out a piteous moan – go on, Harry – higher than that – good – the fur vanished – the fangs shrank – and he turned back into a man.”
- Harry Potter and the Chamber of Secrets, Chapter 10 (The Rogue Bludger)
Then as homework, the students are required to write a poem about Lockhart defeating the werewolf.
“Homework: compose a poem about my defeat of the Wagga Wagga werewolf! Signed copies of Magical Me to the author of the best one!”
- Harry Potter and the Chamber of Secrets, Chapter 10 (The Rogue Bludger)