Generally speaking, you can use your old prescription at hand, but I am not sure if your ADD value is included in your old prescription. If there is ADD value, you don't need to recheck your eyes, but if you don't have ADD value, you have to check your eyes again, Because ADD means near additon which is used for making single vision reading glasses, computer glasses, bifocal & progressive eyeglasses.
No, you don't need a special prescription for computer eyewear as your prescription is not expired. But they may use the different information on your prescription to figure out what strength you need for your computer glasses. You will find there is an ADD number in your prescription. That's what they use to make computer glasses for you.