Students had to identify the man in the picture and say whether they would vote for him or not, using the cartoon "as a source" in their answers.
One answer calling Mr Zuma "too stupid... and selfish with money" to vote for was given two ticks.
The education department for Gauteng province has welcomed the apology.
It says it began investigations into the test question after a complaint from a concerned parent.
The question was part of a Grade Six English Visual Literacy Test at an independent school in the province.
The head of the school has promised to "refrain from using any cartoons that have a potential of causing any form of distress or confusion", the department said in a statement.
The BBC's Milton Nkosi in Johannesburg says the cartoon was drawn in November 2013 by South African cartoonist Jonathan Shapiro, known by his pen name Zapiro.
It related to the controversial security upgrades, which included a swimming pool, worth $23m (£15m) at President Zuma's home in rural KwaZulu Natal.
Last month, the president said he would abide by a court ruling that he must repay part of the money spent on his Nklandla homestead and had never knowingly set out to "violate the constitution".
The shower head in the cartoon is a reference to a statement Mr Zuma made during a trial in 2006 that he had taken a shower after having sex to reduce the chance of contracting HIV. He was acquitted of raping a family friend.
The opposition is pushing for corruption charges against Mr Zuma over an arms deal in 1999 to be reinstated.
Mr Zuma has always denied the allegations - and the charges were dropped just weeks before the 2009 election which led to him becoming president.