If you have a bottling bucket with a spigot, that would work fine. The 9" false bottom should be the right size, but measure first just to make sure. It's pretty easy to connect the false bottom to the spigot. You need a short length (about 3.5") of 1/2" OD plastic tubing, about 2 - 3" of 3/8" OD copper tubing, a couple hose clamps, and a small drilled rubber bung. One end of the plastic tubing fits on the outlet from the false bottom. Clamp it with a hose clamp. The copper tubing fits in the free end of the tubing. Fit the other hose clamp over the the plastic tubing where the copper inserts into it, but don't tighten it yet. The rubber bung fits over the exposed end of the copper tubing with the narrow end pointing outwards. You don't want the copper tubing to protrude out of the bung. The bung then plugs in to the back side of the spigot. Get the false bottom centered in the bucket, and you can tighten the loose hose clamp in the right position.
The only problems I've had with this was that the false bottom can float, allowing grains to get underneath it, stirring the mash too vigorously can pull the rubber bung out of the spigot, and it is susceptible to stuck sparges if you're using a lot of sticky stuff like flaked barley.
To stop the floating, always add foundation water before adding any grains, and drain out a small amount to flood the outlet hose with water. It's then dense enough to stay on the bottom.
To fix the stuck sparges, add enough foundation water to support the grain before adding any grain. Then add some grain, more water, more grain etc until it is all mixed up. Provided the grist is supported by some water, I've never had a stuck sparge.
As for the stirring, be careful. Mixing small amounts of grain and water at a time, makes this much easier.
The reasons I switched to a cooler instead of a bucket were that my bucket was too small, and to provide extra insulation to maintain temperatures during the sparge. I could have got a larger bucket, and insulated it, but the cooler was so much easier.
-a.