Резервне живлення працює дві години.

Because резервне has to agree with живлення in gender, number, and case.

• живлення is a neuter singular noun.
• So the adjective must also be neuter singular: резервне.

A quick comparison:

• резервний акумулятор = backup battery, masculine
• резервна система = backup system, feminine
• резервне живлення = backup power / backup power supply, neuter

In this sentence, живлення means power supply or electrical power.

Depending on context, живлення can mean things related to feeding, nutrition, or power supply. Here, because of the full phrase резервне живлення, it means backup power or backup power supply.

So the sentence is talking about an emergency or reserve source of electricity, not food.

Yes. Працює literally means works or is working.

In Ukrainian, працювати is often used for machines, systems, devices, services, and power sources in the sense of:

• functioning
• operating
• running

So Резервне живлення працює means something like:

• The backup power works
• The backup power is operating
• The backup power runs

All of those are natural translations depending on context.

Because година is a feminine noun.

The numeral two has two main forms in Ukrainian:

• два for masculine and neuter nouns
• дві for feminine nouns

So:

• два дні = two days
• два вікна = two windows
• дві години = two hours

Since година is feminine, дві is required.

After 2, 3, 4, Ukrainian usually uses a special counting pattern:

• дві години
• три години
• чотири години

But after 5 and higher, you normally get годин:

• п’ять годин
• десять годин

So:

• 2 hours = дві години
• 5 hours = п’ять годин

This is one of the standard number + noun patterns in Ukrainian.

It means for two hours.

Ukrainian often expresses duration without a separate word like for. So:

• працює дві години = works for two hours

English needs for, but Ukrainian usually does not.

This is very common with time expressions:

• Я чекав годину. = I waited for an hour.
• Вона спала три години. = She slept for three hours.

It is present tense, third person singular.

The infinitive is працювати = to work.

Conjugation here:

• я працюю = I work / am working
• ти працюєш = you work / are working
• він / вона / воно працює = he / she / it works / is working

Since резервне живлення is singular and treated as it, the verb is працює.

By itself, працює дві години most naturally means works / runs for two hours as a duration statement.

In context, it can suggest:

• its operating time is two hours
• it runs for two hours
• it remains functional for two hours

If you specifically wanted has been working for two hours already, Ukrainian would usually say something like:

• Резервне живлення працює вже дві години.

The word вже helps make that meaning clearer.

If you wanted will work for two hours, you would often use future context:

• Резервне живлення працюватиме дві години.

Because Ukrainian has no articles.

So Резервне живлення can mean:

• backup power
• the backup power
• a backup power supply

The exact choice in English depends on context, but Ukrainian does not mark that distinction with articles.

The normal neutral order here is:

• Резервне живлення працює дві години.

Ukrainian word order is fairly flexible, but changing it can shift emphasis.

For example:

• Дві години працює резервне живлення.
  This emphasizes the duration.
• Працює резервне живлення дві години.
  Less neutral, more context-dependent.

For a learner, the original order is the best standard pattern to remember.

That would be a different structure:

• Резервне живлення працює протягом двох годин.

This also means The backup power works for two hours / over a period of two hours, but it is more explicit and a bit more formal.

Notice the case change:

• дві години = simple duration expression
• протягом двох годин = after протягом, you use genitive

So both are possible, but працює дві години is simpler and very natural.