Für unsere alte Wohnung brauchen wir einen Nachmieter, weil die Kündigungsfrist noch läuft.

German main clauses follow the verb-second rule: the conjugated verb must be in the second position.

So in this sentence:

• Für unsere alte Wohnung = first position
• brauchen = second position
• wir = comes after the verb

That is why you get:

• Für unsere alte Wohnung brauchen wir einen Nachmieter

and not necessarily:

• Wir brauchen für unsere alte Wohnung einen Nachmieter

Both are possible, but starting with Für unsere alte Wohnung puts extra focus on the apartment.

Because für takes the accusative case.

The noun Wohnung is feminine, so in the accusative feminine, the article and adjective forms are:

• unsere alte Wohnung

For feminine nouns, the accusative looks the same as the nominative in this kind of phrase, so:

• nominative: unsere alte Wohnung
• accusative after für: unsere alte Wohnung

If it were dative, you would see forms like unserer alten Wohnung, but that is not used after für.

Not usually. Here alte Wohnung usually means former apartment or previous apartment, not necessarily an apartment that is physically old.

German often uses alt in this way:

• mein alter Chef = my former boss
• meine alte Schule = my old/former school

So in this sentence, unsere alte Wohnung most likely means the apartment we are moving out of.

Because brauchen takes a direct object, and the direct object is in the accusative case.

Nachmieter is masculine, so:

• nominative: ein Nachmieter
• accusative: einen Nachmieter

That is why the sentence says:

• Wir brauchen einen Nachmieter

A Nachmieter is the person who takes over an apartment after the current tenant moves out. A natural English translation is:

• replacement tenant
• new tenant who takes over the lease/apartment

The word is built from:

• nach- = after
• Mieter = tenant

So literally, it is something like next tenant or follow-up tenant.

They are different kinds of tenants:

• Nachmieter = a new tenant who moves in after you leave
• Untermieter = a subtenant, someone renting from the main tenant rather than directly from the landlord

So in this sentence, they do not want a subtenant. They want someone to replace them after they move out.

Because weil introduces a subordinate clause, and in German subordinate clauses, the conjugated verb typically goes to the end.

So:

• main clause: Für unsere alte Wohnung brauchen wir einen Nachmieter
• subordinate clause: weil die Kündigungsfrist noch läuft

Compare:

• Die Kündigungsfrist läuft noch.
• ..., weil die Kündigungsfrist noch läuft.

After weil, the normal word order changes and the verb moves to the end.

Here noch means still.

So:

• die Kündigungsfrist läuft noch = the notice period is still running

It shows that the period has not ended yet.

Common meanings of noch include:

• still
• yet
• another / one more, depending on context

In this sentence, still is the right meaning.

German often uses laufen in a figurative way for things like periods, contracts, deadlines, or processes. It means something like:

• to be running
• to still be in effect
• to continue

So:

• Die Kündigungsfrist läuft noch
  means
• The notice period is still running / still in effect

This is very natural German, even though the literal verb is runs.

Kündigungsfrist is a compound noun:

• Kündigung = termination, cancellation, notice
• Frist = deadline, time limit, period

Together, Kündigungsfrist means:

• notice period
• period of notice required before ending a contract

In housing contexts, it refers to the amount of time between giving notice and actually being allowed to end the rental agreement.

In German, subordinate clauses are normally separated from the main clause by a comma.

Since weil introduces a subordinate clause, you need a comma:

• ..., weil die Kündigungsfrist noch läuft.

This is standard German punctuation.

Usually, yes. In this sentence, brauchen simply means to need:

• Wir brauchen einen Nachmieter = We need a replacement tenant

It is a very common verb and usually takes an accusative object.

It can also appear in other structures, for example:

• Ich brauche Hilfe. = I need help.
• Du brauchst nicht zu kommen. = You do not need to come.

But in your sentence, it is the straightforward need meaning.

Yes. That version is also correct:

• Wir brauchen für unsere alte Wohnung einen Nachmieter, weil die Kündigungsfrist noch läuft.

The difference is mainly focus:

• Für unsere alte Wohnung brauchen wir ... emphasizes the apartment
• Wir brauchen für unsere alte Wohnung ... is a more neutral statement

German is flexible because of case marking, but the verb-second rule still applies in the main clause.