I have a class B license, will I need to take the general and air brake written tests again when I move up to an A?
Depends what state you're in. In Cali, if it's been less than a year since you passed the the Class B written tests, you only have to take the combination test. More than a year, you have to re-do all tests including class c.
Your B license is a provisional license for the A License, therefore You shouldn't have to take any written tests accept those that will be a new endorsement. But to refresh the information wouldn't hurt. You will have to take the Skill Test and provide your own vehicle at MVD or use a 3rd party tester.