With the lower unit removed you aren't pumping water. The water pump is in the lower unit and must be submerged in a drum filled with water or connected to a water hose with an adapter referred to as "Muffs".....a pair of 3-4" rubber cubs, joined by a spring steel bar and containing a female water hose adapter.
Muffs slip over the water intake holes in the lower unit just above the bulge that contains the prop shaft and gears and are available at Walmart in the sporting goods section along with Pennzoil TC-W3 premium 2 cycle oil for mixing 50 parts gas to 1 part oil (50:1) in your gas tank.
Once you get your lower unit with the water pump installed, and with the hose running water...which will spew out around the muffs to some extent, and engine running, water will come out through the prop hub with the exhaust and some "mist" may come out the holes in the rear of the mid section just below the bulge in the alum casting that is formed around the engine proper which is the exhaust relief for use when the engine is in N gear.
A pee source is usually a ⅛" pipe tap into either the exhaust water jacket cover.....flat plate on the Port (left) side looking forward from behind, or a tap into the block water jacket cover...flat plate, across the back of the engine block, through which the spark plugs are mounted. You can add one if you choose. Just drill, tap and install. ⅛-¼ tubing will work for your hose barb sizing.