Czwartego maja po pracy idziemy do muzeum na wystawę.

In Polish, dates are usually expressed with the genitive case.

• czwartego = fourth in the genitive
• maja = May in the genitive

So czwartego maja means on the fourth of May / on May 4th.

A helpful way to think about it is that there is an implied word like dnia (day) in the background:

• (dnia) czwartego maja

Because dzień is masculine, the ordinal takes the masculine genitive singular ending -ego.

Maja is the genitive singular of maj.

Month names in Polish usually change form in dates:

• maj → maja
• czerwiec → czerwca
• lipiec → lipca

So after a day number, the month is normally in the genitive:

• 1 maja
• 4 maja
• 15 maja

In Polish, names of months are normally written with a lowercase letter, unlike in English.

So:

• maj
• czerwiec
• grudzień

This is standard Polish spelling.

Po pracy means after work.

The preposition po often requires the locative case when it means after in time expressions.
That is why praca changes to pracy.

• praca = work
• po pracy = after work

Similar examples:

• po obiedzie = after lunch
• po lekcji = after class

Polish often uses the present tense of an imperfective verb to talk about a planned future action.

So idziemy can mean:

• we go
• we’re going
• we are going to go in a planned sense

This is very natural in Polish, especially with a time expression like czwartego maja.

It works a lot like English:

• On May 4th we’re going to the museum.

Because the verb already tells you the subject.

• idziemy = we go / we’re going

In Polish, subject pronouns are often omitted when they are clear from the verb ending.
You could say my idziemy, but that would usually add emphasis, such as:

• We are going, not someone else.

Because the two prepositions express two slightly different ideas:

• do muzeum = to the museum as a place/building
• na wystawę = to an exhibition as the event or attraction

So the sentence means something like:

• we are going to the museum
• specifically for/to an exhibition

This pattern is very common in Polish:

• iść do kina na film = to go to the cinema for a film
• iść do teatru na spektakl = to go to the theatre for a performance

It actually is in the genitive here, but the form happens to stay the same.

Muzeum is one of those neuter loanwords ending in -um that usually do not change in the singular.

So:

• nominative: muzeum
• genitive: muzeum

That is why after do you still see muzeum, not a different-looking form.

Because na with a verb of motion usually takes the accusative case.

Here the idea is movement toward something:

• na wystawę = to the exhibition

The noun wystawa is feminine, and its accusative singular form is wystawę.

Compare:

• Idziemy na wystawę. = We’re going to the exhibition.
• Jesteśmy na wystawie. = We’re at the exhibition.

The second sentence uses the locative because it describes location, not movement.

No. Polish word order is fairly flexible.

The given order is natural and neutral:

• Czwartego maja po pracy idziemy do muzeum na wystawę.

But other orders are possible, for example:

• Po pracy czwartego maja idziemy do muzeum na wystawę.
• Idziemy czwartego maja po pracy do muzeum na wystawę.

The main meaning stays the same, but the emphasis changes. Polish often moves time or place expressions around for style or focus.

Usually, no.

In a short sentence like this, expressions such as Czwartego maja and po pracy do not normally need commas:

• Czwartego maja po pracy idziemy do muzeum na wystawę.

You might sometimes see commas for a special pause or stylistic reason, but in ordinary writing they are not necessary.

Yes. Both are correct.

• 4 maja = a common numeric way to write the date
• czwartego maja = the same date written out in words

In speech, Polish very often uses the fully declined spoken form:

• czwartego maja

So a written 4 maja is usually read aloud as czwartego maja.