The closest international airport is Jacksonville International Airport (JAX) although it is possible to enter through other cities (e.g. Miami or Orlando) especially if the intent is to hire a car. Jacksonville to St. Augustine is possible through several shuttles which provide door-to-door service or shuttles on a schedule. They can take you directly to the Casa Monica hotel in St. Augustine. You should book in advance by email and you can use major credit cards to do so. Here are several suggestions (there are others): Airport Shuttle of St. Augustine. Door-to-door (2 people) is available from 4:00 AM to Midnight, $75 one way; $5 for each additional person. They also offer a shuttle for $60 initial charge and $5 per person extra up to six. But given how close the price is for door-to-door, that seems the better option. Email them at: firstname.lastname@example.org
Another option is Island Airport Shuttle, whose rates are comparable for door-to-door. They too take major credit cards.
Their email is: email@example.com
They also provide service from Orlando International Airport but that runs to double the price from JAX.
One-way or two-way car rentals are available from all US airports, and provide another option.