المطعم ده هادي بعد الشغل.

In Egyptian Arabic, the demonstrative ده means this for a masculine singular noun, and it usually comes after the noun:

• المطعم ده = this restaurant

That is different from English, where this comes before the noun.
A feminine example would be:

• العربية دي = this car

So the pattern is very commonly:

• noun + ده / دي / دول

Because in Arabic, the verb to be is usually not said in the present tense.

So:

• المطعم ده هادي
  literally looks like
• this restaurant quiet

but it means:

• This restaurant is quiet

This is completely normal in Egyptian Arabic. If you wanted a past meaning, then a form of كان would usually appear.

هادي is the predicate adjective in the sentence. It describes the restaurant and means quiet or calm.

So the sentence breaks down like this:

• المطعم ده = this restaurant
• هادي = quiet
• بعد الشغل = after work

So the full idea is: This restaurant is quiet after work.

هادي is the common Egyptian Arabic form of the adjective.
The more formal / Modern Standard Arabic version is هادئ.

In everyday Egyptian speech, many words are pronounced and written in a more colloquial way, so learners will often see:

• هادي in Egyptian Arabic
• هادئ in Modern Standard Arabic

Both relate to the idea of being quiet/calm, but هادي is what fits this Egyptian sentence.

Yes. Adjectives in Arabic usually agree with the noun they describe in gender and number.

Here:

• مطعم is masculine singular
• so the adjective is also masculine singular: هادي

If the noun were feminine, you would normally use the feminine form:

• القهوة دي هادية = This café is quiet

So هادي matches مطعم.

It means after work or after getting off work.

• بعد = after
• الشغل = work / the job

So بعد الشغل is a very natural way in Egyptian Arabic to say after work.

In this sentence, it tells you when the restaurant is quiet.

Because Arabic often uses the definite article in places where English does not.

So even though English says:

• after work

Egyptian Arabic naturally says:

• بعد الشغل

This does not necessarily mean a very specific piece of work. It is just the normal Arabic way to express the idea.

Although it is written with الـ, the ل is not pronounced normally here, because ش is a sun letter.

So الشغل is pronounced roughly like:

• ish-shoghl or esh-shoghl

not:

• el-shoghl

This is a very common pronunciation rule with the Arabic definite article.

Yes.

• المطعم هادي بعد الشغل = The restaurant is quiet after work
• المطعم ده هادي بعد الشغل = This restaurant is quiet after work

So ده adds the meaning this.

A simple way to see it is:

• المطعم ده = this restaurant
• هادي = quiet
• بعد الشغل = after work

So the order is basically:

• [subject] + [description] + [time phrase]

Literally:

• this restaurant quiet after work

Natural English:

• This restaurant is quiet after work.