Wir brauchen eine andere Lösung für dieses Problem.

Because Lösung is a feminine noun, and here it is the direct object of the verb brauchen.

• die Lösung = the solution
• In the accusative, feminine stays die if definite, or eine if indefinite.

So:

• Wir brauchen eine Lösung.
• Wir brauchen eine andere Lösung.

The adjective andere has the ending -e because it comes after eine and describes a feminine singular noun.

In German, every noun has a grammatical gender: masculine, feminine, or neuter.
Lösung happens to be feminine, so its dictionary form is die Lösung.

This is something you usually need to learn together with the noun:

• die Lösung
• das Problem

Even though solution in English has no grammatical gender, German nouns always do.

Because the preposition für always takes the accusative case.

So after für, you need the accusative form of the article or determiner.

Here, Problem is neuter:

• nominative: dieses Problem
• accusative: dieses Problem
• dative: diesem Problem

Since für requires accusative, the correct form is:

• für dieses Problem

Not:

• für diesem Problem

Because Problem is a neuter singular noun, and for dieser-word determiners, the nominative neuter and accusative neuter forms are both dieses.

So:

• Dieses Problem ist schwierig.
  nominative
• Wir lösen dieses Problem.
  accusative

The form stays the same, even though the case changes.

Andere is an adjective meaning other or different.

So:

• eine Lösung = a solution
• eine andere Lösung = another solution / a different solution

In this sentence, it tells you that the current solution is not the one wanted; a different one is needed.

German main clauses normally follow the verb-second rule, often called V2.

In this sentence:

• Wir = first element
• brauchen = second element
• the rest follows after that

So:

• Wir brauchen eine andere Lösung für dieses Problem.

This is the normal word order for a statement.

You can move another element to the front, but the verb still stays second:

• Für dieses Problem brauchen wir eine andere Lösung.

That version is also correct, but it emphasizes für dieses Problem more.

Yes. That is completely correct.

German often allows different word orders as long as the conjugated verb stays in second position in a main clause.

Compare:

• Wir brauchen eine andere Lösung für dieses Problem.
• Für dieses Problem brauchen wir eine andere Lösung.

Both mean the same basic thing, but the second one gives slightly more focus to for this problem.

Brauchen means to need, while müssen means must / have to.

So:

• Wir brauchen eine andere Lösung. = We need another solution.
• Wir müssen eine andere Lösung finden. = We have to find another solution.

The sentence with brauchen focuses on the need itself.
A sentence with müssen would focus more on obligation or necessity to do something.

Yes. Very often, brauchen is used with a direct object in the accusative:

• Ich brauche Hilfe.
• Wir brauchen Zeit.
• Sie brauchen eine andere Lösung.

That is exactly what happens here:

• eine andere Lösung is the accusative object of brauchen

So the structure is very similar to English need + noun.

Because in German, all nouns are capitalized.

So:

• wir is not capitalized, because it is a pronoun
• brauchen is not capitalized, because it is a verb
• Lösung and Problem are capitalized, because they are nouns

This is one of the most noticeable spelling rules in German.

In this sentence, it functions like an adjective modifying Lösung.

You can think of the noun phrase like this:

• eine = article
• andere = adjective
• Lösung = noun

So andere behaves like an adjective and gets an adjective ending:

• eine andere Lösung

No, that would be incorrect in standard German.

You need an article or some other determiner here:

• Wir brauchen eine andere Lösung.
  correct
• Wir brauchen andere Lösungen.
  also correct, but plural: other solutions
• Wir brauchen andere Lösung.
  incorrect

Without eine, the singular noun phrase does not work properly here.

They are similar, but not identical.

• eine andere Lösung = a different / another solution
• eine neue Lösung = a new solution

Andere contrasts with the current or previous solution.
Neue emphasizes that the solution is new.

Sometimes both ideas overlap, but they are not always the same.

Yes, the sentence could end there:

• Wir brauchen eine andere Lösung.

That is grammatically complete.

The phrase für dieses Problem simply adds more information and tells us what the solution is for.

So:

• Wir brauchen eine andere Lösung.
  complete but less specific
• Wir brauchen eine andere Lösung für dieses Problem.
  complete and more specific