There is a 5:40 p.m. Maya Island flight (US$63 one-way, with possible cash discounts) from the international airport to Caye Caulker. You should be able to make that. The same flight continues on to San Pedro (either destination is the same price).
I'm not sure exactly what you mean by "mainland shores." There really aren't any beaches around Belize City. Well, there is a manmade beach at Old Belize, on the Western Highway, mainly for cruise daytrippers. You can go to the harbourfront or elsewhere and see the water, but the swimming isn't appealing.
You'd have to drive to someplace like Hopkins for a decent beach, and that's a 2 1/2 hour+ trip by car, probably not a good idea to drive it after dark on a first visit to Belize.
Belize City itself isn't a very good introduction to Belize. Most people would advise you head on out to Ambergris or Caulker.