The New Jersey Transit Authority provides several routes that include Shrewsbury as one of their stops.
In the website below, route 831,832, 833, 835 all start in Red bank and flow through Shrewsbury to destinations south and east like Long Branch, Asbury Park or Freehold.
There did not seem to be any bus service directly to Shrewsbury from any other major city in New Jersey (like Trenton or Newark).
So if you want to use the public bus system and you are outside Red Bank/Shrewsbury area then you need to take train or taxi to Red Bank then bus it to Shrewsbury.
The closest airport with commercial flights anywhere near Shrewsbury is the Trenton Mercer Airport in Trenton, NJ. Fairly small airport with local commercial jets and helicopter service.
See the "Transportation" section of the website below for information on train and public bus and taxi options.
Budget and Hertz both have car rental facilities at the air port.