the alternator can be removed without removing that mount but it is much easier with it off. jack up the car and place on jack stands. take the jack and a block of wood and support the engine under the oil pan. remove the 6 18mm nuts and remove the mount. you may have to remove the bracket for the power steering line before removing the mount but it has to be done anyways, an 8mm and 10mm wrench for that. use the same 8mm wrench to disconnect the ground strap bolted to the bracket. using a 10mm 1/4" drive socket and ratchet, reach down the behind the alternator to undo the nut holding the power wire on. also disconnect the large plug by pressing the metal tab and pulling. leave the small single wire plug until you have unbolted the alternator and can see what your doing. go below and remove the 2 10mm bolts holding the splash shield on. using a 13mm wrench, release the tension on the belt and remove the belt from the alternator. then remove the bolts holding the alternator to the bracket. there are 2 from under the car, one facing the passenger side and one facing the drivers side (this one can be extremely difficult to break loose). the third bolt is up above. once all the 3 bolts are out, slide the alternator from the bracket and rotate it so you can see the small plug. using a small flat head screwdriver CAREFULLY pop the plug off, it will break if you are not careful. turn the alternator so you are holding the pulling upwards and the double ears are between the head and the intake manifold. from there you can wiggle it out.
HTH