Will this 22" drawer slide replace the slide mounted to my 18" drawer?
I would use the correct size. Here is an 18" one. The 22" will be to long for the draw. 

If you use the 18" it will fit the draw, but you will likely need to add wood to the side to allow a place to screw the track into.

EDIT:
You need to add a piece of wood to the side of the cabinet. That wood could be glued and clamped until set. Then screw the side track to that piece of wood.

You may be able to add a board stretching from side to side, then screw the end of the 18" track into that. 

I see no easy way unless you find a place that sells that exact slide.

One more thought:
If you used the 22" slide and just fastened the first 18" that came in contact with the draw. It may mean drilling new holes in the slide however if the design is like I think it is, I think that would work. Then you could attach the slide to the front and back as original.