На табло написано, что посадка начинается через двадцать минут.

Good question. Табло is one of the many Russian nouns that are indeclinable, meaning the form does not change in different cases.

So:

• табло = display board / screen / panel
• на табло = on the board / on the screen

Even though на here requires the prepositional case for location, the word табло itself keeps the same form.

This is common with some borrowed nouns ending in -о, such as:

• метро
• кино
• радио
• такси

So на табло is grammatically normal, even though the noun does not visibly change.

Literally, it means something like:

• On the board, it is written
• or Written on the display is...

In natural English, you would usually say:

• The board says that...
• It says on the screen that...
• The display says that...

Russian often uses this impersonal pattern:

• написано = it is written
• сказано = it is said
• указано = it is indicated

So На табло написано... is a very natural way to report written information.

Because this is an impersonal construction.

Написано is the short passive participle in the neuter singular, and in expressions like this it works like English it is written.

There is no specific noun in the sentence agreeing with написано. Russian uses the neuter singular form in many impersonal statements:

• Здесь написано, что... = It says here that...
• В книге сказано, что... = It is said in the book that...

If you were describing an actual noun, then agreement would matter:

• Текст написан = The text is written
• Статья написана = The article is written
• Письмо написано = The letter is written

But in На табло написано, it is just the impersonal it is written pattern.

Here что means that, not what.

It introduces a clause reporting the content of what is written:

• На табло написано, что... = It is written on the board that...

So the sentence structure is:

• На табло написано = It says on the board
• что посадка начинается через двадцать минут = that boarding begins in twenty minutes

This is very common in Russian after verbs or expressions of saying, writing, knowing, noticing, etc.

Yes — посадка is a broad noun with several meanings depending on context.

It can mean:

• planting plants or trees
• landing of an aircraft
• boarding passengers onto a plane, train, etc.

In airport context, посадка usually means boarding.

For example:

• Посадка на рейс открыта. = Boarding for the flight is open.
• Идёт посадка. = Boarding is in progress.

So in this sentence, because of табло and the time announcement, посадка is understood as boarding.

Russian often uses the present tense of an imperfective verb for scheduled or expected future events.

So:

• посадка начинается через двадцать минут literally = boarding begins in twenty minutes

Even though начинается is grammatically present tense, the phrase через двадцать минут makes the future meaning clear.

English does something similar in some cases:

• The movie starts at 8.
• The train leaves in ten minutes.

So this is completely natural Russian.

The verb here is начинаться, which means to begin / to start.

Compare:

• начинать / начать = to begin something
• начинаться / начаться = to begin, to start

So:

• Они начинают посадку. = They begin boarding.
• Посадка начинается. = Boarding begins / Boarding is starting.

The -ся form is used because посадка is the thing that is beginning.

After через in the meaning in or after, Russian uses the accusative case.

So:

• через минуту = in a minute
• через час = in an hour
• через день = in a day
• через двадцать минут = in twenty minutes

With numbers, time expressions can look a little tricky. After двадцать, the noun appears in the form минут, which is the form used with numbers like five, six, twenty, thirty, etc.

Examples:

• через две минуты
• через пять минут
• через двадцать минут

So the whole phrase means in twenty minutes, not through twenty minutes in the physical sense.

They are very different:

• через двадцать минут = in twenty minutes
  This tells you when something will happen from now.

• за двадцать минут = in twenty minutes / within twenty minutes
  This tells you how long it takes to complete something.

Compare:

• Посадка начинается через двадцать минут.
  = Boarding starts in twenty minutes.

• Мы доехали до аэропорта за двадцать минут.
  = We got to the airport in twenty minutes.

So in your sentence, через is the correct choice because it refers to a point in the future.

Yes, Russian word order is flexible, although the original version sounds very natural.

Original:

• На табло написано, что посадка начинается через двадцать минут.

You could also say:

• На табло написано, что через двадцать минут начинается посадка.

This version puts more focus on through/in twenty minutes.

You might also hear:

• Посадка начинается через двадцать минут, как написано на табло.

But that changes the emphasis and sounds a bit less neutral.

In general, the original sentence is a standard, natural informational sentence.

Not in the same neutral sense.

• На табло написано = It says on the board / It is written on the board
• На табло написали = They wrote on the board

Написали suggests an action performed by someone, even if the people are not named. It focuses more on the act of writing.

Написано focuses on the current written information visible there.

So for signs, notices, screens, and boards, написано is usually the better choice.

Russian has no articles like a or the.

So посадка can mean:

• boarding
• the boarding
• sometimes even a boarding, depending on context

In this sentence, the context makes it definite in English:

• boarding starts in twenty minutes

If you translate too mechanically, you may feel something is missing, but in Russian nothing is missing at all. Context does the job that articles do in English.

It is very natural, especially in travel or airport contexts.

It sounds like something you might read in a textbook, announcement, or travel dialogue, but it is also realistic Russian.

Very similar real-life variants would be:

• На табло написано, что посадка начнётся через двадцать минут.
• На табло указано, что посадка начинается через двадцать минут.
• Там написано, что посадка через двадцать минут.

Your sentence is clear, standard, and idiomatic.