Depending on where you are from, the word "apartment" can mean something different to everyone. Where I'm from originally in the U.S., apartments are typically 3-story buildings with vinyl/brick exteriors. The walls are very thin and noise from neighbors is something you live with. In other places in the U.S., apartments are sometimes in high-rise buildings. It's different everywhere. So when you ask about apartments in Belize, there really aren't any as you probably know them.
What Belize DOES have for rent is lots of condos. Some of these are owned by developers and/or associations, but a large number are owned by separate individuals. Again, the word
"condo" means different things depending on where you are from. Before visiting Belize for the first time, I would not have considered renting a condo because of my perception of what they are where I'm from. Many of the people who plan trips to Belize to look at real estate have the same reaction before they have been there.
If you decide to rent something in Belize, there is a lot to choose from. However, most of the condos for rent are either on or near the beach. This is not a bad thing, since the majority of inquires I receive are from people who prefer living on the beach. The photo here shows a typical condo that you would be able to rent on the beach in Belize. These are generally concrete structures built to hurricane standards. Once you step inside, you can feel how well these are built. With concrete you don't really hear people walking above you or any other noise coming from other units.
The price you pay really depends on location, length of stay and time of year. You can rent something for one night, a week, 3 months or a year. It's all very negotiable. But keep in mind - if you are trying to rent something in "high season" in a beach area, the owner is giving up nightly rentals that are much higher than anything he/she could expect from a long-term renter. For this reason it can be difficult to find long-term rentals on the beach that extend through high-season. It’s not impossible, but you have to do some searching. Something that most people don’t know is that the majority of condos in Belize come completely furnished, including beautiful wood furniture made in Belize.
Once you have your feet on the ground in Belize and decide which area you are interested in, what time of year you are looking for and how long you will be there, it's much easier to find an answer to this question.