All I know is that you have hypokalemia in alkalosis because Potassium will shift inside the cell to maintain the electrical neutrality as the hydrogen ions are being pumped out of the cell to offset the alkalosis.
Am not aware if there's any difference between acute or chronic alkalosis and I'd be really interested if someone explains that for us.
Chronic alkalosis ---> hypokalemia (mechanism explained by Mary above)
Acute alkalosis --> hyperkalemia, the mechanism here is not yet fully understood but theories are that it's due to alpha adrenergic stimulation and beta adrenergic block because they found that alpha blockers can offset the hyperkalemia seen in acute alkalosis.
However, for the purpose of USMLE. It's fair enough to understand the mechanism of chronic alkalosis and hypokalemia and no need to bog your brain with the acute alkalotic hyperkalemia.
Also, don't forget that the reverse is true also, meaning that acidosis causes hyperkalemia which is another reason for your DKA hyperkalemia that you mentioned in the other thread.