Use combo moves, spins and swaps to move five numbers, usually one at a time, to their correct balls. The trick is to find combo moves that work together well, preserving mostly the same positions.
It is also probably easiest if the balls you are targeting are next to one another. For instance, if you are trying to get 1 through 5 in place, you can arrange to move 5 to the 1-ball, and then use Right to move it onto the 2-ball at the beginning of the "protected" group of four balls (2 through 5). Moving 5 to the 1-ball should be very easy; if it is on the 0-ball, use Swap; otherwise just spin it around the ring to the 1-ball. Next work on moving 4 to the 1-ball using only moves that leave 2 through 5 undisturbed, and tick it in next to the 5, and so forth.
The trick is finding useful moves like that! Some moves that only protect balls 2 through 4 or 3 through 5 might be helpful, too.
You will probably notices that moves that leave four numbers fixed usually will move the remaining eight number in two cycles of four. Having found one such move, you will probably need a different move that leaves the "protected" four numbers fixed but mixes up the other numbers between those two cycles.