To start with that Viper probably has a lot more power than you BIG TIME.
Your going to have to get at least a H/C setup or put on a supercharger to be able to hang with him. Thats assuming that the Viper has not been modified.
You don't need to swap the transmission for more power. Automatics have advantages over manuals and vice versa. You could change the converter for a higher stall. That will definitely help you with the launching of the car from a stop. A 3.55 or 3.73 rear differential will also help you get an edge without having to crack open the engine. Removing the CATS (catalytic converter) will also help that engine breath easier.
These things will add to the performance and you will get closer to your goal of beating him but your still going to need a little bit more. If you can afford it a supercharger (cost around $5K) kit for that C5 will help out a lot. Vortech just released thier kit for corvettes in particular C5's. Lingenfelter has heads and cam combos for $2.5k if you want to do it yourself. Actually I am asking about a "sleeper" combo that will have excellent daily driven manners and still be able to put Vipers away
That will provide you with what you need to beat him. If you do enjoy while you can because he is going to get right back at you
Frank