Ihr neuer Lippenstift passt gut zur Bluse, aber ohne Handcreme werden ihre Hände im Winter sofort trocken.

By context, not by the capital letter alone.

At the start of a sentence, German capitalizes the first word automatically, so Ihr could in principle be:

• her
• their
• formal your

If the meaning shown to you is her new lipstick, then this Ihr is just her. The capital I is only there because it begins the sentence.

A useful extra point:

• Ihr with a capital I in the middle of a sentence usually means formal your
• ihr with a lowercase i can mean her, their, or to her, depending on context

Because Lippenstift is masculine singular nominative, and ihr works like an ein-word determiner.

In ihr neuer Lippenstift:

• ihr = possessive determiner
• Lippenstift = masculine singular noun
• the whole phrase is the subject, so it is in the nominative

With masculine nominative singular, the possessive word often has no visible ending, so the adjective shows the ending instead:

• ihr neuer Lippenstift

Compare:

• ihre neue Bluse — feminine
• ihr neues Kleid — neuter
• ihre neuen Schuhe — plural

passen zu is a very common combination meaning:

• to go well with
• to suit
• to match

So passt gut zur Bluse means that the lipstick goes well with the blouse.

Why zur?

• zu takes the dative
• Bluse is feminine
• dative feminine article = der

So:

• zu der Bluse → zur Bluse

This contraction is very common and natural.

Because Handcreme is being used in a general, substance-like sense: without hand cream.

German often leaves out the article with nouns like this when speaking generally, especially with materials, substances, or products used as an uncountable idea.

Also, ohne takes the accusative, but here you do not see that clearly because there is no article.

Compare:

• ohne Handcreme = without hand cream in general
• ohne eine Handcreme = without a hand cream / without any hand cream
• ohne die Handcreme = without the specific hand cream

Because German main clauses use the verb-second rule.

In the second clause:

• aber introduces the clause
• ohne Handcreme is placed in the first position
• the finite verb werden must come second
• the subject ihre Hände comes after the verb

So the structure is:

aber | ohne Handcreme | werden | ihre Hände ...

You could also say:

• ..., aber ihre Hände werden ohne Handcreme im Winter sofort trocken.

That is also grammatical. The version in your sentence puts more emphasis on without hand cream.

No. Aber is a coordinating conjunction, so it does not send the verb to the end.

That means the clause after aber still keeps normal main-clause word order:

• aber ohne Handcreme werden ihre Hände ...

Compare that with weil, which is a subordinating conjunction:

• ..., weil ihre Hände im Winter sofort trocken werden.

There, the verb werden goes to the end.

Because werden + adjective means to become / to get.

So:

• trocken werden = to become dry / to get dry

This is different from:

• trocken sein = to be dry

So:

• ihre Hände werden trocken = her hands become dry
• ihre Hände sind trocken = her hands are dry

German could also use a verb like austrocknen in some contexts, but trocken werden is a very normal and clear way to express this idea.

Im is a contraction of in dem.

So:

• im Winter = in the winter / in winter

German often uses im with seasons and other time expressions:

• im Winter
• im Sommer
• im Januar

This is a standard and very common contraction.

Because all nouns are capitalized in German.

That is one of the most visible spelling rules in German:

• der Lippenstift
• die Bluse
• die Handcreme
• die Hände
• der Winter

This is different from English, where common nouns are normally lowercase.

A useful detail:

• Ihr at the beginning is capitalized because it starts the sentence
• the nouns are capitalized because they are nouns