Na terça-feira, minha prima vai ao teatro comigo.

Na is the contraction of em + a.

• em can mean on / in / at, depending on context.
• terça-feira is feminine, so it can take the article a.
• So em + a terça-feira becomes na terça-feira.

In natural English, this is usually just on Tuesday.

Also, in Brazilian Portuguese, you may sometimes hear just terça-feira with no preposition, especially in informal speech:

• Terça-feira, minha prima vai ao teatro comigo.

Both are possible, but na terça-feira is very clear and standard.

The names for Monday through Friday in Portuguese are written with -feira and a hyphen:

• segunda-feira
• terça-feira
• quarta-feira
• quinta-feira
• sexta-feira

This is simply the standard spelling. You should learn them as complete words.
Notice that sábado and domingo do not use this pattern.

In Portuguese, days of the week are normally not capitalized unless they begin a sentence.

So:

• na terça-feira = correct
• na Terça-Feira = not standard

This is different from English, where Tuesday must be capitalized.

The comma is natural here because Na terça-feira is an introductory time expression.

So this is very normal:

• Na terça-feira, minha prima vai ao teatro comigo.

If the time phrase is short, some writers may omit the comma:

• Na terça-feira minha prima vai ao teatro comigo.

Both can be understood, but the comma helps separate the opening time phrase from the rest of the sentence.

In Brazilian Portuguese, both forms are possible:

• minha prima
• a minha prima

The definite article a before a possessive is often optional in Brazilian Portuguese.
So both can mean my cousin.

There is usually no important difference in meaning here.
In this sentence, minha prima sounds completely natural.

Yes.

Portuguese distinguishes between:

• prima = female cousin
• primo = male cousin

English usually just says cousin, so this is one of those places where Portuguese gives more information than English does.

Because Portuguese often uses the present tense to talk about a future event when the time is already clear from context.

Here, na terça-feira makes the future meaning clear, so vai works naturally.

Literally, vai is the present tense of ir:

• ela vai = she goes / she is going

But with a future time expression, it can mean:

• she is going
• she will go

This is very common in both Portuguese and English:

• On Tuesday, my cousin is going to the theater with me.

Because the verb ir normally takes the preposition a when expressing destination in standard Portuguese.

• ir a = to go to

And teatro is masculine and uses the article o:

• o teatro

So:

• a + o = ao

That gives:

• vai ao teatro

So ao is just a contraction:

• ao = a + o

Because in standard Portuguese, after ir, the usual destination pattern is:

• ir a
• or ir para

So:

• ir ao teatro = standard
• ir para o teatro = also possible

By contrast, no teatro usually means in/at the theater, not to the theater:

• Ela está no teatro. = She is at the theater.

You may hear ir no teatro in some informal Brazilian speech, but learners are usually taught to use ir ao teatro or ir para o teatro.

Yes, a little.

Both can mean to the theater, but they are not always used in exactly the same way.

• ir ao teatro is the classic and very idiomatic way to say go to the theater, especially as an activity or outing
• ir para o teatro can sound more like head toward the theater or emphasize the destination

In this sentence, vai ao teatro is the most natural choice.

After com, Portuguese uses special forms for some pronouns:

• comigo = with me
• contigo = with you
• consigo = with himself/herself/yourself (formal)

So:

• comigo = correct standard form

That is why the sentence says:

• ... comigo = ... with me

A learner should treat comigo as the normal word for with me.

Yes. Portuguese word order is somewhat flexible.

These are all possible:

• Na terça-feira, minha prima vai ao teatro comigo.
• Minha prima vai ao teatro comigo na terça-feira.
• Minha prima vai comigo ao teatro na terça-feira.

The original version puts the time first, which gives it a nice frame-setting effect: first the listener knows when, then what happens.

Yes, but only if the subject is already clear from context.

In the original sentence, minha prima is already the subject, so adding ela would be unnecessary.

• Minha prima vai ao teatro comigo. = natural
• Ela vai ao teatro comigo. = also natural if everyone already knows who she is

Portuguese often leaves out subject pronouns when they are not needed, but here the noun minha prima already does the job clearly.