Oh man, the alternator on a '90 Integra. That's one of the crueler jokes that car can tell. I'd like to get my hands on... I mean, have some words with the engineer that came up with that one.
Try un-bolting the driver's side half-shaft carrier bearing frome the back of the engine. It's the black, loop looking thing that your driveshaft goes through. Since the driveshaft is still connected to the trans and the wheel assembly, you can rotate the carrier bearing around it. This should free up some space and allow for a slight displacement of the driveshaft. It gave me enough space to get my alternator out. DON"T force anything. I ain't paying for your screw up.
*This could spare you the task of removing the whole driveshaft. Which is the proper course of action.