1 You cost will vary depending on the distance of your move, but one-bedroom apartment moves usually take between three to five hours, putting your move at about $300 to $500 for labor alone.2  You may also be charged extra fees for special services and supplies.3  Some examples include: 10' U-Haul (suggested for a small apartment move) will cost a flat fee of $19.95 for the day plus $0.99 for every mile traveled. You may also be required to purchase fuel, as some rental companies ask you to return the truck with a half tank of gas.