هدف الادخار مهم لصندوق الطوارئ.

This is a nominal sentence in Arabic. In the present tense, Arabic usually does not use a separate word for is/am/are.

So:

هدف الادخار مهم لصندوق الطوارئ

literally works like:

The savings goal — important for the emergency fund.

Arabic understands this as The savings goal is important for the emergency fund.

Because هدف الادخار is an iḍāfa construction, often called a genitive construct.

In an iḍāfa:

• the first noun usually does not take ال
• the whole phrase becomes definite or indefinite based on the second noun

So:

• هدف = goal
• الادخار = the saving / the savings

Together:

• هدف الادخار = the savings goal or the goal of saving

Even though هدف has no ال, the whole phrase is definite because الادخار is definite.

Because the first noun in an iḍāfa construction does not take tanwīn.

Normally, an indefinite noun might look like:

• هدفٌ

But in iḍāfa, it becomes:

• هدفُ الادخارِ

So the missing tanwīn is a clue that هدف is linked to the noun after it.

The same thing happens in:

• صندوق الطوارئ not
• صندوقٌ الطوارئ

الادخار is a verbal noun (called a maṣdar) from the verb ادّخر meaning to save, to put aside, or to set aside money.

So الادخار can mean:

• saving
• savings
• the act of saving money

In this sentence, it functions as a noun inside the iḍāfa:
هدف الادخار = savings goal / goal of saving.

Because مهم here is the predicate of the sentence, not an adjective directly attached to a noun.

Compare:

• الهدف المهم = the important goal
  Here المهم is an adjective describing الهدف, so it matches in definiteness.

But in your sentence:

• هدف الادخار مهم = The savings goal is important

Here مهم is the khabar (predicate), and predicates in this kind of sentence are typically indefinite, so مهم is correct, not المهم.

Because it matches the grammatical head of the subject, which is هدف.

• هدف is masculine singular
• so the predicate is مهم

If the subject were feminine, you would expect:

• مهمة

For example:

• الخطة مهمة = The plan is important

In your sentence, the main noun is هدف, so مهم is the correct form.

The preposition لِـ often means:

• for
• to
• for the benefit of
• sometimes with respect to

Here it means for:

• لصندوق الطوارئ = for the emergency fund

So the sentence is saying that this savings goal is important for an emergency fund.

Also, after لِـ, the following noun is in the genitive case.

Yes, exactly.

• صندوق = fund / box
• الطوارئ = emergencies / emergency situations

Together:

• صندوق الطوارئ = the emergency fund

Just like هدف الادخار, this is an iḍāfa, so:

• the first noun (صندوق) does not take ال
• the whole phrase is definite because the second noun (الطوارئ) is definite

Literally, it is something like fund of emergencies, but in natural English we say emergency fund.

Arabic and English do not always package ideas the same way.

الطوارئ is a noun referring to emergencies or emergency situations, and Arabic commonly uses it in expressions like:

• قسم الطوارئ = emergency department
• حالة طوارئ = state of emergency
• صندوق الطوارئ = emergency fund

So even though English uses emergency like an adjective, Arabic often uses a noun meaning emergencies inside an iḍāfa.

Fully vocalized, it would be:

هَدَفُ الادِّخارِ مُهِمٌّ لِصُنْدُوقِ الطَّوارِئِ

Here is why:

• هَدَفُ → nominative, because it is the subject (mubtadaʾ)
• الادِّخارِ → genitive, because it is the second term of an iḍāfa
• مُهِمٌّ → nominative, because it is the predicate (khabar)
• لِصُنْدُوقِ → genitive after the preposition لِ
• الطَّوارِئِ → genitive, because it is the second term of another iḍāfa

In normal Arabic writing, these endings are usually not written.

Yes. That is a very natural literal reading.

The iḍāfa هدف الادخار can be understood as:

• the goal of saving
• the savings goal

Which English version sounds best depends on context. In financial language, savings goal is often the most natural translation, but grammatically the Arabic structure is literally goal of saving/savings.

Yes, Arabic word order is fairly flexible.

The basic order here is:

• subject: هدف الادخار
• predicate: مهم لصندوق الطوارئ

But for emphasis, Arabic can move phrases around, for example:

لصندوق الطوارئ هدف الادخار مهم

This is less neutral and more marked, but it is possible in the right context. The original sentence is the most straightforward textbook-style order.