For all the things you love about Kitty, his litter box probably isn't one of them. It can be unsightly to have a box a cat poop sitting around your house. You may have opted for a covered cat box to make his potty less of an eyesore. This may not be the best option for keeping you and kitty happy.
Undercover Litter Box
The most obvious advantage of a covered cat box is that you don't see what's inside. Most covers are opaque, and some even have a little flap for Kitty to use as a door. Some are designed to specifically not look like a litter box at all. While this is ideal for you, it could be off-putting for Kitty. If you don't see the mess, you're less likely to clean the box every day. Covered boxes hold on to smells more than open ones. If the box is too stinky, Kitty may go on strike. Poop on the floor is even more of an eyesore than what you were initially trying to prevent.
If you have a single kitty and he's a bit shy, he may appreciate the privacy a covered box offers. You'll want to give him some space to himself anyway, and a covered box is an easy way to do that, especially if you live in a small apartment. If you've only got one cat, or possibly two, then even if you go a day without scooping, the smell shouldn't get too offensive. If you have multiple cats, make sure each kitty has his own box.
One drawback to a covered box, is that it has an obvious opening. This can give a place your dominant cat to stalk and ambush your new kitten or a shy kitty whose using the box. Your dominant cat will have a door to guard. He may also poop outside the box to mark his territory. This will be a sign to other cats in the home that he's in control. Using an open box will make it harder for one kitty to corner another and your dominant fuzzball won't have a door to guard.
Daily cleaning of all the litter boxes in your home will be a sure way to keep odors to a minimum. Each week the boxes should completely dumped of litter and soaked in hot water to eliminate odors building up over time. For stubborn stains, you can add a gentle dish soap to the water. Avoid using air fresheners with strong aromas near his box, because Kitty may avoid it altogether if he's not a fan of the scent. A layer of baking soda will absorb unpleasant odors but won't repel Kitty. Make sure whenever you scoop his box, replace the litter you removed with fresh litter. A clean litter box is more important to Kitty than a pretty one.