shouldn't need to spend too much money on a SLAM solution. and yes SLAM works
outdoors
Last I checked you could get an xbox 360 kinect used for 25 a pop at gamestop. As long as whatever you are running can run ros on ubuntu you should be good. Odroid makes good cheap single board computers, and while I don't have any experience with the jetson boards, they should also be rather good, if not a bit on the expensive side going by what I've read on CD.