eAPIS needs to filed outbound, but you don't need to call anyone outbound.
Typically, it's not too hard to find an internet connection, even internationally, so getting your eAPIS filed shortly before departure typically isn't all that hard. As far as calls are concerned, I made my best guess oin time, added 20 minutes to it, and that usually has me there spot on (it always takes longer than you think). However I've never called en-route. Once I was even off by about 45 minutes, and the guy didn't care at all. He was getting paid overtime to sit around for me to show up.
Going to Mexico, you don't have to call them. Just file your flight plan and take off. Seemed weird to me being used to Canada and the US, but that's how it is.
I don't know if you'll have a handler or not (or where you're going), but in Cozumel, after landing you need to close your flight plan as well as go through customs. First customs (don't be alarmed by the guys with large guns who walk up to greet you), then closing your flight plan involves stamps from 4 different offices. Make sure to have your aircraft registration, airworthiness, and proof of Mexican insurance when you go through. The Comandancia (police) will want to see it. The guy about tore the original airworthiness cert for the Aztec, so if you're flying an older plane, be aware. After you pay the taxes for entry with your plane, make sure they give you the paperwork that allows the plane to be brought in anytime the rest of the year without paying that penalty.
Also, there is no immigration or emmigration tax for crew. When Laurie and I go we're both required crew on the Aztec/310.
Have fun! I really enjoy flying to Mexico.