"Centrally located" in Belize would mean Belize City and environs, but not many visitors would want to get an apartment there, due to high prices and safety issues. You might want to look at a Belize map and see what options you have, such as San Ignacio, Hopkins and Belmopan.
It's nearly impossible to arrange an apartment or house rental before you come, except possibly on Ambergris Caye where there are condos and other rentals catering to tourists. Best to do that in person after you arrive -- you'll save money and headaches by seeing what you are getting into before you pay money.
Belize books and publications by Lan Sluder:
Belize First Magazine (http://www.belizefirst.com
Fodor's Belize & Guatemala Guide (new 4th edition coming this summer)
Adapter Kit: Belize (the first and only comprehensive guide to living, retiring, working and investing in Belize -- a best-seller on Amazon.com)
San Pedro Cool (the first and only visitor guide to Ambergris Caye)
Belize First Guide to Mainland Belize
UpClose Central America
Belize Book of Lists