Pani Mario, spotkanie mamy czwartego kwietnia o dziewiątej rano.

Because this is the vocative form, used when directly addressing someone.

• Pani Maria = Ms Maria / Mrs Maria as the basic dictionary form
• Pani Mario = Ms Maria! when speaking to her

In Polish, names often change form when you are talking to the person.

Also note:

• Pani stays the same
• only Maria changes to Mario

So the comma after Pani Mario makes sense too: it marks direct address.

The comma separates the form of direct address from the rest of the sentence.

So:

• Pani Mario, ... = Ms Maria, ...

This works much like English:

• Maria, the meeting is on...
• Sir, please wait
• Anna, can you help?

In Polish, when you directly address someone, that part is usually set off with commas.

Yes, literally spotkanie mamy means we have a meeting, but in Polish this is also a very natural way to say that a meeting is scheduled.

So here mamy does not necessarily stress possession. It often means something like:

• we have the meeting on...
• the meeting is scheduled for...
• our meeting is on...

This is common everyday Polish.

Yes. Both are possible, but the word order changes the emphasis a little.

• Mamy spotkanie czwartego kwietnia... = a more neutral We have a meeting on April 4th...
• Spotkanie mamy czwartego kwietnia... = puts spotkanie first, as the topic: As for the meeting, we have it on April 4th...

Polish word order is more flexible than English, so fronting a word often gives it extra focus or makes it the topic of the sentence.

Because in Polish, the verb ending already shows the subject.

• mamy = we have

So my is usually unnecessary unless you want emphasis or contrast.

Compare:

• Mamy spotkanie jutro. = We have a meeting tomorrow.
• My mamy spotkanie jutro, a oni pojutrze. = We have a meeting tomorrow, and they have theirs the day after tomorrow.

English normally needs the subject pronoun, but Polish often drops it.

Because Polish dates are normally expressed with an ordinal number plus the month in the genitive case.

So:

• czwartego = fourth in the required case
• kwietnia = of April / April in the genitive

This is the standard pattern for dates in Polish:

• pierwszego maja = the first of May
• dwudziestego lipca = the twentieth of July

A helpful way to think about it is that an omitted word like dnia is understood:

• czwartego kwietnia ≈ on the fourth day of April

Because Polish often gives dates without a preposition in this kind of sentence.

So English says:

• on the fourth of April

But Polish commonly says simply:

• czwartego kwietnia

This is normal and idiomatic.

Polish does use prepositions with some time expressions, but not all:

• w poniedziałek = on Monday
• o dziewiątej = at nine
• czwartego kwietnia = on the fourth of April without an extra preposition

Because after the preposition o when telling time, Polish uses the locative case.

So:

• o dziewiątej = at nine

There is also an implied noun:

• o dziewiątej godzinie = at the ninth hour

Since godzina is feminine, the numeral/adjective form is feminine too:

• dziewiąta = basic form
• o dziewiątej = locative form after o

Other examples:

• o pierwszej = at one
• o drugiej = at two
• o siódmej = at seven

Often o dziewiątej is enough, but rano makes it completely clear that it is 9 a.m.

• o dziewiątej = at nine
• o dziewiątej rano = at nine in the morning

This is especially useful when the time could otherwise be ambiguous.

Common time-of-day words include:

• rano = in the morning
• po południu = in the afternoon
• wieczorem = in the evening
• w nocy = at night

Yes, it is normal. Pani Mario is polite, but less distant than using a surname.

It is common in workplaces, schools, offices, and many semi-formal situations.

Compare:

• Pani Mario = polite, but somewhat personal
• Pani Kowalska = more formal, using surname
• Mario = informal, used with someone you are on ty terms with

So this sentence sounds polite and natural.

Yes. Polish often allows shorter, more natural versions, but you can make it more explicit.

For example:

• Pani Mario, mamy spotkanie czwartego kwietnia o dziewiątej rano.
• Pani Mario, spotkanie mamy czwartego kwietnia o godzinie dziewiątej rano.

Adding o godzinie is more explicit, but in everyday speech it is often omitted because o dziewiątej already clearly means at nine o’clock.

So the original sentence is concise and natural.