Alternator replacement on this car with a zetec is a PITA. I have an 01 SE. If I can remember correctly you have to do the following.
Move the engine coolant resivoir. I disconnected the bolts and clip(towards the back of the engine bay) and moved it to the front.
Move the Power Steering Fluid reservoir. This just lifts up and pulls out. The three rubber grommets keep it in place.
Next you have to move the wires and fuel rail supply. The fuel rail supply uses a special tool. However, if you use some intuition you can use a pop bottle cork.
Now, once everything is disconnected you can push the engine as forward as it will go. The service guide says to drop the alternator down through the bottom, but if you position it correctly you can get it in from the top. Dropping it down is easier when taking it out.
I would make sure that it is your alternator and nothing something else in the charging circuit. I believe my voltage regulator or something else is dead/on the way out. If you know someone at a professional shop they can test it for you. If not you can pay to get them tested.