Integra's can be really good for lap racing, not for drag. For the reason being that you need to invest quiet some cash for it to be running mid 13's. I would recommend if you have enough cash invest in a acura integra type r, a bit hard to find but you can find one. Or you can also just find yourself a clean gsr that you can work on. Just make sure it has vtec. An LS would not be too bad either. You can do turbo or LS/VTEC swap. An old acquaintance of mine had a full stripped civic with an LS B18 motor, gsr tranny, aftermarket intake maifold, and throttle body, intake, full racing headers, running on open exhaust, cammed, chipped ecu, and on slicks and it was running low 13's.
I would just say find yourself a good running gsr. You can always upgrade the valve train. You can strip the car and get carbon fiber exterior parts, but, I do not think you can lower the weight down below 2,000 lbs. And muscle cars are just in another level, another tier per se. A lot of investment if you wish to keep up with gas guzzlers. Not impossible tho, but money can make a lot of things possible. Good luck on your decision, and make it wise.