Assuming you mean Antigua, the island in the West Indies, and not Antigua, Guatemala...
The cheapest place to fly to might be Cancun. It's not too difficult to use the Mexican ADO express buses to get to Chetumal, then get one of the water taxis to Ambergris Caye.
The bus from Cancun to Chetumal takes about 6 hours and costs a few hundred pesos, but take your time and visit Playa del Carmen or Tulum. Check the bus schedules & fares here:
https://www.ticketbus.com.mxThe water taxis depart Chetumal daily in the afternoon; about 90 minutes and US$25 to $35.
More details about how I do the trip are on my BelizeBus site:
http://belizebus.wordpress.com/It has links to the water taxi services too.
If you find a cheap flight to Honduras, there are weekly water taxis from Puerto Cortes to Belize.
If you find a cheap flight to Guatemala City, there are daily express buses to Flores, then change to Belize City. Details are on BelizeBus.