Mow the grass as short as you can and de-thatch the area (remove excess organic material) with a garden or leaf rake.
Water the de-thatched soil to soften it for easier digging. A product such as Nutrasoft can also be used to soften clay soils.
Dig your holes just a little larger than the size of the plug. Plugs are not very big--according to Zoysiafarms.com, they are about 1 inch--and a good hand spade will work. Dig holes at least 12 inches apart. Keep in mind that zoysia grass will eventually take over your lawn, so you don't want it planted too close.
Add a light coating of fertilizer at the bottom of the hole. A few pellets of fertilizer will do the job.
Place the plug in the hole and fill it back in with the soil that was taken out.
Press firmly on and around the plug to be sure there is no air pocket underneath or on the sides.
Water the newly planted area thoroughly. Water again daily for at least a week or two or until the plugs are well established. Do not over-water after the plugs are established.