Remove the cactus, carefully, from its existing pot. A cactus that has sharp spines can be wrapped in several layers of newspaper for handling without damage to the plant or you. If your cactus is root-bound to the point you cannot safely extract the plant without damaging the roots or cactus, you may need to carefully break the pot from around the roots.
Gently brush and remove the old soil from around the cactus roots with your fingers. A part of re-potting a cactus is to freshen the soil medium to provide more nutrients to the roots. Removing as much of the old soil as possible allows more of the new soil and nutrients to the roots.
Place a shard of a broken pot over the drain hole of the new pot, then put 1 to 2 inches of fresh cactus soil in the bottom of the pot, over the shard. The shard should not be a tight fit over the hole, but is placed there to allow water excess drainage while retaining some soil moisture.
Set your cactus on top of the soil in the bottom of the new pot. Because you have selected a pot that is only slightly larger than the old pot, your cactus should be planted at the same depth as it was in the old pot.
Fill the pot and around the cactus with the fresh, new potting soil. Gently pat down the soil surface to eliminate any air pockets and to ensure there is good root contact with the soil medium.
Place your re-potted cactus away from direct sun and do not water it for one to two weeks to allow any root damage to heal. Watering your newly potted cactus immediately after re-potting may cause damaged roots to develop bacteria infections or fungus.