Excluding vocabulary items which are entirely new or have fallen into disuse, what are some ways in which Japanese syntax itself has changed between the 1950s and today?
(I would also like to exclude phenomena such as the semi-productivity of terms like ググる, since I'd argue these are new vocabulary and not fully productive ways of deriving new words.)
言文一致 and the shift to modern kana orthography (1946) happened shortly before this period, and they were undoubtedly much more fundamental than anything that happened after 1950. But I'll focus on the change of modern 口語 here. The list is far from complete; I just wrote down things that came to mind.
Decline of ましょう as a way to express future inference.
明日は雨が降りましょう。
This was common in weather forecast until the 1970's, but we never hear this today. See: https://ameblo.jp/heppokomental/entry-12527905042.html
Decline of some "heavy" keigo patterns (でございます, でありました, くださいませ, ...)
Continued decline of ぬ as a negation marker (ありませぬ, 分からぬ, ...)
Decline of many gender-specific sentence endings (かしら, だわ, ...), although many remain in fictional works as part of お嬢様言葉
Decline of most iteration marks (ゝ, 〱, ...) except 々. (If I understand correctly, these symbols have never been officially standardized nor banned by the government.)
Increased acceptance of ら抜き of ichidan potential forms (食べれる, 見れる, ...)
Increased acceptance of i-adjective + です (嬉しいです, よかったです, ...) and the decline of ～うございます. See: Conjugating present and past negative i-adjectives
If you mean by "syntax" basic structures of the language and fundamental rules of how you construct a sentence (like the subject-object-verb order), I'd say nothing has changed. Most languages don't change at the fundamental level in 70 years, or in the lifetime of a person.
That said, which grammatical words and constructions are preferred have changed in some cases, especially in spoken Japanese. Off the top of my head, in negation constructions, ないです is a much more acceptable alternative to ありません than it was, and -ぬ (as in 足らぬ) became less preferred over -ない (as in 足りない). You hear less いかに and いかなる for "how" and "what", more どう/どのように and どんな. I'm sure there are many more examples like these. The old words like these can still be spoken, and maybe part of set phrases in some cases, but they generally add an old-fashioned tone to how you speak. (Not too unlike how "thou" and "thee" may sound in English, I think.)
There is a famous song starting with 兎追いしかの山. This would be 兎を追ったあの山 ("the mountain where I chased after rabbits") in today's Japanese. Children learn this song in school and may misunderstand 追いし as 美味し(い), because the 追いし form of the verb 追う is simply not used any more.
