The phrase 'within a day' is correct and can be used in written English. You can use it to signify that something will happen in a 24-hour period. For example, "I will have the report finished within a day.".
But within a day he was vomiting.
Most died within a day or so.
In 2007, he came within a day.
Within a day, most people are up and running again.
Within a day, the sore had dried up.
Within a day, she said, 200 had signed up.
Within a day, scientific surveys found big wins for Obama.
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union