She accuses the newspaper of damaging her reputation in a 2017 editorial linking her to the 2011 shooting of US Representative Gabby Giffords.
Her positive test was announced by Judge Jed Rakoff in New York on Monday.
The judge said the trial will be now begin on 3 February.
"She is of course unvaccinated," the judge noted earlier, referring to Ms Palin.
Ms Palin, a former governor of Alaska, sued the Times in 2017, saying it damaged her reputation with an opinion piece written by its editorial board which said that her political rhetoric helped to incite the 2011 shooting in Arizona.
It also said Ms Palin's political action committee circulated a map of electoral districts that put Ms Giffords and 19 other Democrats under "stylised crosshairs".
The newspaper conceded the initial wording of the editorial was flawed and quickly corrected it.
Ms Giffords was shot in the head and severely wounded, but survived the attack. Six people were killed and another 13 wounded.
Ms Palin, the 2008 Republican vice-presidential nominee alongside Arizona Senator John McCain, has had Covid before.
She has previously said "it will be over my dead body that I'll have to get a shot".
In 2016, she backed Donald Trump's bid to win the Republican nomination for the US presidency.