Who applies an airworthiness directive when the airframe is newer than the AD?
I am interpreting your question as asking whether an owner of an aircraft is responsible for complying with ADs that were issued before the aircraft was manufactured.

Short answer: yes.

There are several types of ADs that affect your aircraft. Some of them apply to all aircraft of a specific type, some apply to specific serial numbers or model numbers, and some apply to specific parts that are installed on your aircraft.

Some are one-time and others are recurring. The one-time items would usually have been checked before the airplane left the factory. The exception would be if they were required after a certain number of airframe hours. Recurring items can still apply to all aircraft—even those made after the AD was issued. 

The two ADs listed below require the owner to check for excessive wear in the listed parts. I just replaced the Bendix switch in my Cherokee because it failed the required test in the AD.

Here’s one that was issued in 2011 for all Cessna aircraft AD 2011-10-09:


  We are superseding an existing airworthiness directive (AD) for Cessna
  Aircraft Company (Cessna) 150, 152, 170, 172, 175, 177, 180, 182, 185,
  188, 190, 195, 206, 207, 210, T303, 336, and 337 series airplanes.
  That AD currently requires repetitive inspections and replacement of
  parts, if necessary, of the seat rail and seat rail holes;


Here’s one that was issued in 1976 that applies to all aircraft that have Bendix ignition switches AD 76-07-12: 


  Applies to all aircraft employing magnetos and using Bendix ignition
  switches listed in the table below except switches identified by four
  digit date code (new) adjacent to the model number or a white dot
  (modified) on the support plate adjacent to the Bendix logo.


Your A&amp;P should have software that allows them to easily check which ADs apply to your aircraft and then create an AD Compliance Report for you. It will list all of the ADs and the date that they were complied with. Most reports also list ADs that apply to your aircraft model, but are excluded by serial number.