Yes. It's quite simple, actually, and should't take more than 10 minutes if it's your first time.
You'll need a 10mm socket, an 8mm socket, a phillips head screwdriver, and a flathead screwdriver.
There are two 10mm bolts holding the bumper on under the front of the car. Remove them.
There are also two phillips screws on the front far sides of the bumper. Remove them.
Inside the wheel well, at the tip of the bumper, there are two 8mm bolts holding the bumper up. Using an 8mm socket and an extension [so you don't chip the paint with the wrench], remove them.
Finally, pop your hood. Directly above your grill there about seven snaps holding your bumper to the front of the car. use the flathead screwdriver to gently pry up the lock piece from each one and remove it. Then gently remove the seat of the snaps.
Once all these are removed, gently rock your bumper off. It will come straight out and away from the car. If it seems stuck on the ends, gently pull the end pieces down and away from the fender area and the wheel well. Sometimes it catches on the place where you removed the 8mm bolts.
When the bumper comes off, turn it over onto a towel or something to prevent it from scratching, and remove the grill. Assembly is the reverse of removal.
(Note: this is how it is for my 2-door EX. If it is a slightly different for your 4-door LX, I apologize if there is any confusion. Here's a diagram for your specific model year to help you identify the parts
http://www.hondaautomotiveparts.com/auto/jsp/mws/prddisplay.jsp?inputstate=5&catcgry1=CIVIC&catcgry2=1999&catcgry3=4DR+LX&catcgry4=KA5MT&catcgry5=BUMPER