Mam nadzieję, że ta bateria wytrzyma do wieczora.

Because mieć nadzieję is the standard Polish expression for to hope.

• mam = I have
• nadzieję = hope (in the correct case here)

So Polish says literally I have hope, where English usually says I hope.

You can conjugate it normally:

• mam nadzieję = I hope
• masz nadzieję = you hope
• mamy nadzieję = we hope

Polish often uses a noun phrase where English uses a verb.

Because after mieć in this expression, the noun goes in the accusative case.

• dictionary form: nadzieja = hope
• accusative singular: nadzieję

So:

• mam nadzieję = I have hope / I hope

This is a very common pattern in Polish: after mam (I have), the thing you have is usually in the accusative.

Że means that and introduces a subordinate clause.

So the structure is:

• Mam nadzieję = I hope
• że ta bateria wytrzyma do wieczora = that this battery will last until the evening

In English, that is often optional:

• I hope this battery lasts until the evening
• I hope that this battery lasts until the evening

In Polish, że is very natural and usually expected in this kind of sentence.

Because bateria is a feminine singular noun, so the demonstrative this must match it.

• masculine: ten
• feminine: ta
• neuter: to

Since bateria is feminine:

• ta bateria = this battery

This is grammatical agreement: the demonstrative must match the noun in gender, number, and case.

Here wytrzyma means will last, will hold out, or will endure.

The verb wytrzymać often suggests that something keeps going successfully despite limits, pressure, or time. With a battery, it means the battery will keep working long enough.

So in this sentence:

• ta bateria wytrzyma do wieczora = this battery will last until evening

It can also be used in other contexts:

• Nie wytrzymam! = I can’t stand it / I won’t endure it
• Most wytrzyma duży ciężar. = The bridge will تحمل / withstand a heavy load

Because wytrzyma is the future form of the perfective verb wytrzymać.

In Polish, many verbs come in aspect pairs:

• wytrzymywać = imperfective
• wytrzymać = perfective

Here the speaker is talking about a single complete result in the future: whether the battery will last all the way until evening. That is why Polish uses the perfective future:

• wytrzyma = it will last / it will hold out

By contrast, wytrzymuje is present tense and would mean something like it lasts / it is enduring, which does not fit this sentence.

Because Polish often expresses the future without a separate word like English will.

With perfective verbs, the present-looking form usually has future meaning.

So:

• wytrzyma looks like a present-tense form in shape
• but because wytrzymać is perfective, wytrzyma means will last

This is a very important feature of Polish aspect.

Because the preposition do requires the genitive case.

• dictionary form: wieczór = evening
• genitive singular: wieczora

So:

• do wieczora = until evening / until the evening

This is a fixed grammar rule:

• do domu = to home / homeward
• do pracy = to work
• do rana = until morning

It usually means until evening or until this evening, depending on context.

In natural English, this sentence is often translated as:

• I hope this battery lasts until this evening
• I hope this battery lasts until tonight

The exact English wording depends on the situation, but the Polish phrase itself is straightforward:

• do wieczora = until evening

Because Polish often omits subject pronouns when they are already clear from the verb ending.

• mam already means I have
• so ja is unnecessary in a neutral sentence

That is why:

• Mam nadzieję... is the normal version

You can add ja for emphasis or contrast:

• Ja mam nadzieję, ale on nie. = I hope so, but he doesn’t.

Yes. Polish word order is more flexible than English, though not completely free.

The neutral order here is:

• Mam nadzieję, że ta bateria wytrzyma do wieczora.

But other orders are possible for emphasis, for example:

• Mam nadzieję, że do wieczora ta bateria wytrzyma.
• Ta bateria mam nadzieję, że wytrzyma do wieczora.
  This is possible, but more marked and less neutral.

So the original sentence is a very natural, standard way to say it.

Usually yes, especially for everyday batteries and also device batteries in general speech.

• bateria = battery
• akumulator = rechargeable battery / accumulator / car battery, depending on context

In everyday Polish, people often say bateria even for a phone battery:

• Bateria mi pada. = My battery is dying.

So in your sentence, ta bateria sounds completely natural.

A rough pronunciation guide is:

Mam nadzieję, że ta bateria wytrzyma do wieczora.
mahm nah-JEY-yeh, zheh tah bah-te-RY-a vih-TSHI-mah doh vye-CHO-rah

A few useful sound notes:

• ż sounds like the s in measure
• rz here also sounds like ż
• wy is roughly vih
• cz sounds like ch in chop
• ę at the end of nadzieję is often pronounced more lightly in normal speech

This is only an approximation, but it can help you get started.