Learn German: Das Fahrrad von meinem Bruder ist alt.

The preposition von always requires the dative case. Since Bruder is a masculine noun, the possessive word mein takes the masculine dative ending -em to become meinem.

Doesn't von usually mean "from"? Why is it used for "my brother's" here?

You are right that von often means "from" (like coming from a place). However, in everyday German, von is also the most common way to show possession, meaning "of". So das Fahrrad von meinem Bruder literally translates to "the bicycle of my brother".

Is there another way to say "my brother's bicycle" without using von?

Yes! You can use the genitive case (the grammar case for possession) and say das Fahrrad meines Bruders. The genitive is more common in written or formal German, while von with the dative case is what you will hear most often in everyday spoken German.

Why doesn't the adjective alt have an ending, like altes?

Adjectives only take an ending when they come directly in front of a noun (for example, ein altes Fahrrad). When the adjective comes after a verb like ist to simply describe the subject, it is called a predicative adjective and stays in its basic, unchanged form.