1. I went to the video arcade 'yesterday night'. (Is this expression right?)

2. He went to his academy after school to learn English and math.

3. From her first diary, what can we know? Summarize her first diary.

4. Of whomself did she feel proud?
(Is this sentence grammatical?)

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Are all the expressions above grammatical?J

In sentence 1, it's better to phrase it this way:

I went to the video arcade last night. (We don't use the phrasing "yesterday night.")

Sentences 2 and 3 are fine.

Sentence 4 is incorrect. Try this:
Of whom did she feel proud?
(The suffix -self is used only on a pronoun referring to the subject. Here are some examples: I saw myself in the mirror. They believed themselves to be right in their choices. The boys heard themselves over the loudspeaker.)