The adaptor you need will depend on the brand of brakes you have, Avids for example have alignment washers between the adaptor and calliper body so the adaptor is a little shorter than others.
As to disc size it's down to weight and riding style a 140 mm rear disc will lock up a wheel off road or on but when you're using it to slow without lockup the pad/disc heat will build up and heat the fluid in the system this can cause brake fade, a larger disc can get rid of the heat quicker before it heats the brake fluid.
Unfortunately the only adaptors I have are for 140 mm discs (probably why they're spare!) but to be honest the disc in the image looks more like a 160 have you checked the diameter? I've got a 160 mm on my Ruckus but I'm sure it's capable of running a 180 or possibly bigger, the image shows a 160 mm disc and there's plenty of clearance for a bigger one.