The grease I use is a commercial application for vehicle door hinges. It is called "Cheetah" door hinge lube. It is a high pressure waterproof penetrating grease that works really well. I use it for my trailer winch gears, anchor pulleys, my biggest secret OAR LOCKS!!! and what it was designed for- vehicle hinges and check straps. The nicest thing about this type of grease is that it won't drip or run all over. It is runny when you first spray it but dries to a tacky sort of consistency.
You can get the same type of product from any autobody supply store, the brand doesn't matter.
As far as the diameter of the hole, I just sized up the extended spray nozzle with my drill index. 1/8" maybe?? But, I am certain that the pulley DOES NOT have to be removed to drill this hole. The aluminum is soft enough and there is enough slop in the fit that if you are carefull you can free hand this without damaging the spindle. If your pulley is like mine they are mounted on a peened axle- kind of like a rivet and you will never get them reassembled. By the way, if you mess one up somehow, replacements are available at Auburn Sports and Marine for a couple bucks each. So you have nothing to loose!
So there a go... everything I know about maintaining anchor pulleys!