Managing this on public transportation would be pretty difficult, there’s no good way to do this route. I’ve never taken the shuttle but I’m pretty sure either that or an Uber or Lyft ride is your best bet. And give yourself lots of time, the traffic is a killer!
La Guardia is not on any subway line so public transport is difficult. A taxi into the city from LaGuardia is 30+ dollars so I am guessing the shuttle is the best option.