Learn Italian: Il proprietario vuole l'affitto per il mese prossimo. - The landlord wants the rent for next month.

In Italian, time expressions using prossimo (next) or scorso (last) usually require a definite article. You literally say "the month next" (il mese prossimo) or "the week last" (la settimana scorsa).

Could I put prossimo before the noun, like il prossimo mese?

Yes, you can! Both il mese prossimo and il prossimo mese are correct and mean the exact same thing. Putting prossimo after the noun is just a bit more common in everyday speech.

Is proprietario the specific word for "landlord"?

Proprietario literally means "owner", so it refers to whoever owns the property, vehicle, or business. If you want to be very specific about a residential landlord, you might also hear the phrase padrone di casa (master of the house).

This lesson is about possessive pronouns, but there isn't one here. How would I say "his rent"?

To say "his rent", you would use the possessive adjective and say il suo affitto. However, Italian often drops the possessive and just uses the definite article (l'affitto) when it is already obvious from the context who it belongs to or whose responsibility it is.