I'll try to list the grammatical similarities between Japanese & Mandarin below as I find them. Whether these similarities are because Japanese has copied far too much of Chinese, or whether because they're actually related languages way back in time, I dunno.

1. Mandarin uses 的 where Japanese uses の, な, い. In の's case, this includes as a connector for phrases within sentences and not just showing possession for nouns (ex. 行く の は 嫌 だ "I don't like going/I don't wanna go", Mandarin acts in the same way). For example:
　私 の＝我 の　"my"（same 我 as in わがくに　我が国 "my/our country"）
　可愛 い＝可愛 的 "cute"
　小さい の＝小的 "a cute one"

2. Mandarin uses 的 where Japanese uses んだ (=のだ), なの and の. This is described in Japanese lessons as when you're "telling or asking someone for information", and in Chinese lessons as when you're "putting emphasis on information":

　你 是 什么 时候 来 中文 的？ = あなた は いつ 中国 に 来た の？ = "So when did you come to China?"

　我 是 昨天 来 中国 的 = 私 は 昨日 中国 に 来た ん だ = "Well, I came to China yesterday"

Japanese words like 歴史的 "historical" (歴史 = history) are directly borrowing the Chinese 的.

3. Mandarin uses 吗 (么) "ma" where Japanese uses か "ka" I think. This basically means "yes or no?". Note how similar the sounds "ma" and ka" are...
　中国 か？中国 なの？＝中国 吗？

4. Mandarin duplicates words in order to mean plural or "all of them" (same as Japanese: 人 human, 人人 / 人々 everyone, humans).

5. Mandarin apparently uses 呀 "ya" where Japanese uses おや、あら、よ (basically "oh! oh my!" and "hey"). Again note the similar sounds: oya, [y]ara, yo. Pretty sure Japanese dialects say "ya" instead of "yo" also.

6. Mandarin omits personal pronouns and at times particles in the same way as Japanese ("I ate —> Ate").

7. Mandarin has the word 了 "luh" which is used to mean like "now/not anymore (as opposed to before)", and can at times denote past tense. For example "I'm here now" (= as opposed to before when I wasn't here), "That's expensive now" (= After you told me that price I suddenly thought, that's too expensive). I can't quite make a connection yet but the Japanese past-tense isn't actually past-tense, it gets used for all kinds of present-tense stuff (ex. 変わった "is different/strange", 長い首をした "has a long neck"). I'm still in an early stage of learning but 了 might actually be close in usage to the Japanese "yo"...

8. Which tones and sounds were in the Chinese words obviously have a fairly regular pattern to how the pronunciation ended up in Japanese. Japanese basically smoothed out the dipthongs, ex. "tien" became "ten" or "shuo" became "sho/shuu" etc, and depending on the tone it turned into a different sound sometimes in Japanese. In some cases, the Mandarin pronunciation is actually the same as its spelling in the OLD SPELLING SYSTEM of Japanese, which I won't get into here.

9. It's not just "Chinese pronunciation"s of kanji that are the same in Japanese and Mandarin. For example, "flame" (炎) in the so-called Japanese pronunciation is "honoo" ほのお, but in Mandarin the word for "fire" (火) is "huo" which sounds like "hoa". The Japanese for 火 "fire" is "hi". I've seen a few more examples of "Japanese" words being almost the same as "Chinese" but can't remember them right now.

10. Mandarin and Japanese both use "different" words to say "not" when referring to the word "is, exists". The normal Mandarin word for "no(t)" is 不, but before 有 (Japanese ある, 有る) it changes to 没 and you can say 没 alone (no 没有). In Japanese, you have 有る "aru" which changes to 無い "nai" despite that you can actually say "aranai".

11. Mandarin uses a different word for "is" (很) when using an adjective versus the "is" (是) when using a noun. Likewise, Japanese has い、な to mean "is" with adjectives (い can change tense and な was derived from a present-tense verb).

12. The Japanese は "wa" is used at the same times as the Chinese 是 "is" and 呢 "what about...?". Just like 是, は isn't used right after an adjective (we don't say 赤い は in Japanese, and don't say 是 赤 in Chinese).

我 是 中国 人 = 私 は 中国 人 だ "I'm a Chinese person"
谢谢，你 呢？ = ありがとう、あなた は？ "Thanks, and you?"

13. Showing comparision (superlatives etc) is exactly the same:
　JP: 我 より 汝 (は) 大きい "I from/than You (am) big" = I'm bigger than you
　CN: 我 比 你 (很) 大 "I compare/than You (am) big" = I'm bigger than you

14. Various small constructions in Chinese are copied in Japanese, ex. CN 难(難) 吃 "hard to eat" and JP 食べ にくい or 難い "hard to eat". However this includes phrases, such as CN 难 看 "hard to see = ugly" and JP 見 にくい (but properly spelled 醜い) "hard to see = ugly".

15. The construction of chained verbs in Chinese is copied in Japanese by way of the Japanese て-form:

听 不懂 "listen not understand"
= 聞いて 分からなかった "when listening didn't understand"

16. In both languages, emphasis changes depending on word order. Japanese also does this with the difference between "wa" and "ga" instead, but noticably Japanese ends up switching from 私 は to は 私 just as Chinese switches from 我 是 to 是 我:
我 是 昨天 打 你 的 "I hit you yesterday"
是 我 昨天 打 你 的 "It's ME who hit you yesterday"

私 は 昨日 あなた を 打った ん だ "I hit you yesterday"
私 が 昨日 あなた を 打った ん だ "It's ME who hit you yesterday"
昨日 あなた を 打った の は 私 だ "Yesterday, the one who hit you was ME"

17. In Mandarin (不) and Japanese (じゃ ない、じゃ ん), "not, n't" is used in a phrase like "why don't you just (go do x) then?":

点菜 不就 好 了 "why don't you order, then?"
注文 したら いい じゃん "why don't your order, then?" / "wouldn't it be good to order, then?"

However this extremely common usage of the Japanese じゃん usually isn't mentioned in textbooks or lessons; according to Domino Chinese the same is for the Chinese 不.

18. The Chinese 哎呀 "aiya" and 哎哟 "aiyo" are used the same as the Japanese あら "ara". Basically an exclamation of surprize, like "...Huh? I can't find my wallet!"

More later as I find them!