Can I get training in the USA with a PPL from another country, but without converting it?
Part 1: Can a foreigner with a PPL in another country's system receive training in the US?

All other normal prerequisites such as ASFP, medicals, etc. aside, yes, a foreign license holder can receive training in the U.S.

Remember: You do not need a pilot certificate to receive training, but you would need one to solo or take a check ride (such as a student certificate).

If a person holds a foreign certificate, I believe they have to be flying an aircraft registered in their home nation in order to qualify to solo or otherwise act as PIC in a foreign country without obtaining a local certificate.

So, receive training?  Sure.  Solo/PIC/check-ride in an N-registered? No.

If what you mean to ask is "can he obtain a new cert/rating without converting".  No because the FAA does not issue South Korean pilot certificates, so that would not be possible.  The dual training received in the U.S. would certainly be able to be logged and count as valid training back home.

Part 2: ASFP Training Categories

From the flightschoolcandidates.gov FAQ section:


  What category do I qualify for?
  Candidates seeking flight training are separated into one of four
  different categories. A brief explanation of the categories follows:
  
  Category 1 - Candidates who seek flight training in the operation of
  aircraft with a maximum certificated takeoff weight (MTOW) greater
  than 12,500 pounds, but who do not fall into Category 2.
  
  Category 2 - Candidates who seek flight training in the operation of
  aircraft with a maximum certificated takeoff weight (MTOW) greater
  than 12,500 pounds, and who:
  
  Are employed by a foreign air carrier that operates under 14 CFR part
  1546; Have unescorted access authority to a secured area of an airport
  under U.S.C 44936(a)(1)(A)(ii),49 CFR 1542.229; Are a flight crew
  member who has successfully completed a criminal history records check
  in accordance with 49 CFR 1544.230; or Hold an airman's certificate
  that is recognized by the FAA or appropriate US military agency, with
  a type rating for a multi-engine aircraft that has a certificated
  takeoff weight of 12,500 pounds or more.
  
  Category 3 - Candidates who seek flight training in the operation of
  aircraft with a maximum certificated takeoff weight of 12,500 pounds
  or less for the following training events: 
  
  Single Engine Land (SEL)
  -- Includes initial Airman's Certificate, including a private, recreational, or sport pilot certificate. If a private and/or
  commercial license is the candidate's initial FAA license, it is
  considered an initial airman's certificate and is not exempt.
  Instrument Rating (IR) Multi Engine Land (MEL) Each of these training
  events requires a separate training request. Clarification regarding
  Category 3 exemptions is covered in the next Frequently Asked
  Question.
  
  Category 4 - Candidates who seek recurrent training in the operation
  of aircraft with a maximum certificated takeoff weight (MTOW) greater
  than 12,500 pounds, and are current and qualified on the aircraft for
  which they are requesting training. These training requests are
  submitted by the flight training providers -- Candidates need to
  "Select Providers for Recurrent Training" from the AFSP Candidate
  account. If you have further questions regarding your category, please
  contact your Flight Training Provider.


If he wants training intended for an instrument rating in an aircraft under 12,500 Lbs MTOW then category 3.