In Harvest Moon DS Cute, you can obtain a horse after you have shipped 1000 items via your Shipping Bin. The next morning after this task has been completed, Takakura will stop by and deliver your horse. You can use that horse as a portable Shipping Bin, and even ride it around the valley.
To care for your horse, you can use the brush - which is available from Salibara the Blacksmith's for 800G - to pet it for affection. Your Horse cannot enter in any competitions.