First, most info you need is here:
http://www.ka24development.com/index.html - big diff compatibility writeup.
Second, notice that this thread is over four years old.
Third, no offense but most of that info is wrong. Sigh.
The J30 diff will bolt to the S13, but you must use the S13 rear cover (not just the gasket, the gaskets are the same). The axles may or may not be different. Some J30s had five-bolt axles, some had six. ALL S13s in the US had six, so if your diff has five you need the J30 axles.
Next, the ABS issue. All diff casings with ABS are longer than those without. You can probably bolt it up and drive away but eventually (sooner rather than later) the carrier bearing on the driveshaft will fail. Easiest solution is to get a driveshaft from a car with ABS, which is correspondingly shorter. I have no idea what it would mean to "shave" the front of the casing, I can't see how that would work at all; you'd probably just end up with a worthless, wrecked diff.
Last idea - if the J30 diff has six bolt outputs, you can swap the guts and the outputs over to your S13 stock diff casing. You MUST swap the outputs over also, the output shafts for a VLSD are not the same as those for an open diff. You'd also have to swap the ring gears (but not the pinions) - the S13 and J30 don't have the same final drive ratio. In other words, stock S13 casing + stock S13 ring gear + J30 VLSD guts + J30 six-bolt outputs = not having to worry about the driveshaft. This will only work if you have six bolt outputs, or if you've got the five-bolt outputs AND the corresponding axles.
Go read the S15 diff install in the tech section, it talks more about the ring gear and such.