Is zakaah due on rent that has just been received or rent paid when one year has passed?

No zakaah is due on properties that are prepared for rental;
rather zakaah is due on the rent that is paid on them, subject to two
conditions: 

1-That it reaches the minimum
threshold at which zakaah becomes due (nisaab)

2-That one full hijri year has
passed (since acquiring it).

The year starts from when you sign the contract, whether the
rent is paid in advance at the beginning of the year or at the end of the
year. If it was received at the beginning of the year and the year has
passed, then zakaah must be paid on it, or on what is left of it if some has
been spent and some is left. 

If it is received at the end of the year then zakaah must be
paid on it, because one year has passed since the contract was signed. 

Ibn Qudaamah (may Allaah have mercy on him) said in
al-Mughni (4/271): 

If a person rents out
his house for two years for forty dinars, the rent is considered to have
come into his possession at the time of the contract, and he has to pay
zakaah on the whole amount when one year has passed. End quote. 

Shaykh Ibn Baaz was asked about a man who rented out some
property and received payment for one year in advance, and he used it to pay
off some debts – does he have to pay zakaah on this rent? 

He replied: No zakaah is due on rent in such a case, where it
was received from the renter in advance and he used it to pay off a debt,
because a full year did not pass since it came into his possession. What
counts here is the date of the rental contact until the end of the year. If
he received the rent before the end of the year and used it to pay off a
debt or he spent it on things needed for the house, then no zakaah is due on
it. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz,
14/177 

Shaykh Ibn ‘Uthaymeen
was asked about properties that are prepared for rent – is zakaah due on
them? 

He replied: 

No zakaah is due on these properties, because the Prophet
(peace and blessings of Allaah be upon him) said: “The Muslim does not
have to pay zakaah on his slave or his horse.” Rather zakaah is due on the
rent, if one full year has passed since the contract was drawn up. 

For example: a man rents
out his house for ten thousand, and he receives ten thousand after one year
has passed; he has to pay zakaah on this ten thousand, because one year has
passed since the contract was drawn up. Another man rented out his house for
ten thousand – five thousand of which he received at the time of signing the
contract, and which spent within two months, and five thousand to be paid
halfway through the year, which he took and spent within two months. When
the year is over, he has nothing left of the rent, so no zakaah is due on
it, because it did not spend one full year in his possession, and zakaah is
only due if it has spent one year in his possession. End quote. 

Fataawa al-Shaykh Ibn ‘Uthaymeen,
18/208 

And Allaah knows
best.