Get a scope with adjustable objective, so you can get the parallax set for the short Appleseed range (25m).
Install sling swivels and a sling and you're good to go. The gun right out of the box is accurate enough to clean the Appleseed short course. Oh, you might want to pick up two or three extra Ruger factory mags, that will same you some hustling on the line.
Some will recommend an aftermarket sear or sear/trigger. IMHO this is counterproductive for Appleseed because you will end up with a trigger pull much better than any real-world battle rifle. Learn to shoot the factory 10/22 trigger and you'll easily be able to shoot any AR, AK, M14, etc you might want to use when the SHTF.
I'm not sure but I don't think a scope is the way to go with an Appleseed 10/22...
I think you need tech sites and extended 10 round mags.
(these are the two best suggestions I've heard of so far)
22WRM kind of defeats the purpose of using a 22 at an appleseed, which is to use less expensive ammo.
And Snake is right about the trigger pull.. you don't want a combat rifle to have too soft and light of a trigger pull.... bad things can happen to your buddies in front of you.... enough bad things are going on anyway to add a round to the back of the head is not good.
This reminds me, I need to check on the Appleseed projects in my neck of the woods for the next year.... I had hoped to make one this year... but things didn't work out.
I agree but the OP asked specifically about a scope so that's the way I went. I'd much prefer to shoot the Appleseed with good peep sights than a scope--you'll learn more--but keep in mind that there are people whose eyesight just won't let them use irons of any kind anymore.