You should be able to call an Uber/Lyft/Revel directly to the front entrance of the TWA Hotel for pickup, but it'll be about a 15 minute, very roundabout ride to get to T4 (could be longer if traffic is bad, which is frequently the case due to construction).

The AirTrain is free, doesn't require getting in/out of a car, and will probably be faster (including walking) unless you just missed the last train. To get to the AirTrain, walk out the front of the hotel, cross the access road on the crosswalk, follow the shaded walkway to the AirTrain station, and take the elevator up to the platform. I think the total walking distance is about 500ft (the walk from the curb/AirTrain at T4 to your gate will probably be longer). As dhammer53 said, be sure to take the inner loop train that only serves the terminals (which is one stop to T4) and NOT the outer loop train toward Federal Circle/Howard Beach/Jamaica (which goes the other way).

Here's a map with the walking route highlighted: