He has some money in a bank account from which he pays his monthly expenses – does he have to pay zakaah on it?

If the
amount of money reaches the minimum threshold (nisaab) – which is the
equivalent of 85 grams of gold or 595 grams of silver – and a full hijri
year has passed since acquiring it, then zakaah must be paid on it, even if
something is taken from it each month, because of the general meaning of the
evidence that zakaah must be paid on wealth that is subject to zakaah, when
one year has passed. 

If we assume
that this money reached the nisaab at the beginning of the year, then it
fell below the nisaab during the year, no zakaah is due on it, and the
reckoning of a new year should begin when it reaches the nisaab again. 

For example:
if the amount of money reaches the nisaab in Ramadaan, then it falls below
the nisaab in Dhu’l-Hijjah, then you acquire more money which brings it back
up to the nisaab in Rabee’, then you should start reckoning a new year from
Rabee’. 

Al-Nawawi
(may Allaah have mercy on him) said: Our view and that of Maalik, Ahmad and
the majority is that is in order for zakaah to be obligatory on wealth that
is subject to zakaah and for which one year must have passed, such as gold,
silver and livestock, the nisaab should have been present throughout the
year. If it falls below the nisaab for a moment during the year, the year is
cancelled out, and if it reaches the nisaab again after that, a new year
should be calculated from the moment when the nisaab was reached again. End
quote. 

Al-Majmoo’ (5/506) 

And Allaah knows
best.