setting a dc motor to work for a fixed amount of time
There are two possibilities:
Easy way (but with limitations)
I see a delay of 200 (ms)… So if you want to have it working for a minute, than use 60,000.
If 60,000 is not accepted (maybe the input argument is a signed integer, having a max value of 32,767), than use a for loop and do a delay of a second for 60 times.
This solution will make it not respond within those 60 seconds to another left or right command.
Update: See comment of JRobert, 60,000 can be used without a problem.
Prevent a delay
If you want to respond to a left/right command WHILE the motor moving in these 60 seconds, you have to store the time (with millis) in a variable, and check against the time to know when to switch off the motor.
See command of Duncan C. This is very similar to the 'Blink without Delay' sketch which can be found here.
(Actually, there is a third way using interrupts, but I assume the second solution is best in your case, and is responsive enough considering the tasks you want to do with the Arduino.)