Learn Spanish (Latin America): El hombre trabaja muchas horas.

You would use cuántas. The question word for 'how many' has to match the gender and number of the noun it asks about. Since horas (hours) is feminine and plural, you use cuántas: ¿Cuántas horas trabaja el hombre?

Why is it muchas horas and not muchos horas?

In Spanish, words that describe amounts have to match the gender and number of the noun they belong to. Because hora (hour) is feminine and we are talking about more than one, you must use the feminine plural form muchas.

Could I say El hombre trabaja mucho horas instead?

No, because when 'a lot' or 'many' comes right before a noun, it must match that noun, becoming muchas. You only use the fixed word mucho when it describes an action. For example, if you remove the word horas, you can say El hombre trabaja mucho (The man works a lot).

Why does the verb trabaja end in an 'a' here?

Trabaja comes from the regular verb trabajar (to work). Because the subject is el hombre (the man, which equals 'he'), you drop the -ar and add -a. This is the standard rule for regular -ar verbs when talking about 'he', 'she', or 'you (formal)'.