В программке было написано, что после первой части будет антракт.

Here программка means a program leaflet / printed program / playbill, not a computer program.

It is a diminutive form of программа, so it often suggests a small printed program booklet you might get at a concert, theater, or other event.

So in this sentence, В программке было написано... means something like In the program booklet, it was written...

Because here в means in, referring to location, so it takes the prepositional case.

• в программке = in the program booklet
• в программку would suggest movement into something, which does not fit here

Compare:

• В программке было написано... = It was written in the program
• Положи билет в программку. = Put the ticket into the program

So программке is the prepositional singular form of программка.

Было написано is an impersonal passive construction meaning it was written.

Breakdown:

• было = past tense of быть
• написано = short passive participle of написать

Russian often uses this kind of structure when English would say:

• it said
• it was written
• there was written

There is no explicit subject here. Russian does not need a word like English it.

So:

• В программке было написано, что... = In the program, it was written that...

Because this sentence is impersonal.

When a short passive participle agrees with a specific noun, it changes form:

• текст написан
• статья написана
• письмо написано
• слова написаны

But here there is no actual grammatical subject being described. Russian uses the neuter singular short form by default in impersonal constructions:

• было написано
• было сказано
• было сделано

So написано does not agree with программке. It is just the standard form used in this kind of expression.

Что introduces a subordinate clause, like English that.

So the structure is:

• В программке было написано = In the program it was written
• что после первой части будет антракт = that after the first part there would be an intermission

Russian uses что very commonly after verbs or expressions of saying, writing, knowing, thinking, etc.

Examples:

• Он сказал, что придёт. = He said that he would come.
• Я знаю, что это правда. = I know that this is true.

После requires the genitive case.

So:

• после
  • genitive = after

That is why you get:

• первой from первая
• части from часть

Together:

• после первой части = after the first part

This is a very common pattern:

• после урока = after the lesson
• после фильма = after the film
• после концерта = after the concert

Because первой is the genitive singular feminine form of первая.

It has to match части, because adjectives and ordinal numbers in Russian agree with the noun in:

• gender
• number
• case

Since часть is:

• feminine
• singular
• here in the genitive case after после

the ordinal must also be feminine singular genitive:

• первая часть = the first part
• после первой части = after the first part

Because антракт is the subject-like noun in the clause будет антракт = there will be an intermission.

In Russian, with быть in the sense of there is / there will be, the noun is usually in the nominative case:

• Будет концерт. = There will be a concert.
• Был дождь. = There was rain.
• Будет антракт. = There will be an intermission.

Антрактом would be instrumental and would only appear in a different kind of sentence, for example one involving to be in the sense of to serve as or in certain predicative contexts, not here.

This is because of sequence of tense in English, not because the Russian grammar is different here.

Russian says:

• было написано, что ... будет антракт

Literally: it was written that ... there will be an intermission

But English often shifts the future in a past reporting frame:

• It said that there would be an intermission

Russian usually keeps будет because from the point of view of the writing, the intermission is still in the future.

So:

• Russian: past reporting verb + future in subordinate clause
• English: often past reporting verb + would

Both express the same idea.

Антракт is a normal word for an intermission, especially in theater, opera, ballet, or concerts with sections.

Another common word is перерыв, which is more general and can mean break in many contexts.

The difference is roughly:

• антракт = theatrical / performance intermission
• перерыв = general break, pause, recess

So in a concert or theater program, антракт sounds very natural.

Yes, Russian word order is flexible, and other orders are possible. But this order is very natural because it starts with the place where the information was found.

• В программке было написано, что... = neutral, natural
• Было написано в программке, что... = also possible, but less natural in many contexts
• Что после первой части будет антракт, было написано в программке. = possible, but more marked or formal

Russian often puts known or contextual information earlier and new information later. Here, starting with В программке neatly sets the scene: in the program booklet...

Yes, but it would sound a little different.

• В программке было написано... = it was written in the program...
• В программке написали... = they wrote in the program...

The second version implies unnamed people wrote it. Russian often uses third person plural to mean people / they in an indefinite way.

The passive impersonal было написано is more neutral and closer to English it was written or it said, so it is especially natural in this sentence.