In order to get to Tarrytown by train you'll have to get into Manhattan from LaGuardia first and take Metro North from Grand Central Station to Tarrytown....about 30-40 minutes....but there will be waiting time since you have to get to Grand Central first, and then wait for the next train that stops in Tarrytown....more often during the evening rush...less often other times.
Taking a cab from LaGuardia to Tarrytown would cost you the Earth. A limo might be less....but I couldn't say by how much.
I believe flying into Westchester Airport would be faster and cheaper, if that's an option from where you're starting out.
Contact the hotel and ask if they have pick-up service at either/or the Tarrytown train station and the Westchester airport....that might make the decision easier!