Of course you can. You could even program your robot to not need the joystick at all and do everything itself during driver control as far as rules are concerned. Only during the autonomous period is code using input from the joystick forbidden.
I’d be cautious about spending a significant stretch of time in an autonomous routine though, as unless your robot is well-coded and/or movement across the playing field is quite small, you could mess up other robots, the scoring objects, and/or the field and not be able to immediately return to actual driver control to defuse the situation before lasting damage is done.
Specifically on timers, your robot’s Cortex should have 4 general-purpose on-board timers, plus a few specialty timers like nSysTime and nPgmTime. Have one of those timers set to zero at the very beginning of the driver control task, and your robot should be able to accurately tell how much time’s left.