Phoning them up and booking a flight...?
If your main purpose is not flying training but holiday, and if you're going to be flying much less than 18 hours a week, it's hard to see how the INS would require you to obtain an M-1 visa. Further, if you are not training to obtain a license or rating, but you're just going to have a few sightseeing flights with a commercial pilot (doesn't even have to be an instructor if you don't want those hours to count in your logbook as instruction hours) I don't think you'd need TSA clearance as well.
Having said that, I would not automatically assume that you cannot fly solo (or even with passengers) in the US on an NPPL. It is true that there are no ICAO rules for this, so it won't be an automatic and relatively painless process as with a JAR-FCL or UK PPL. But that doesn't mean that the US cannot grant you some sort of restricted PPL based on your NPPL at all.
In the Netherlands we have a "Recreational PPL", which is more or less comparable to the NPPL in the sense that it is only valid within the Netherlands and requires less experience hours than a PPL. When we go on a club trip to a foreign country, we typically ask the country where we're going to for a few waivers. One of those is for RPL holders, assuming they meet all the Dutch requirements with regards to restrictions, medical, currency etc., to fly in that foreign country. These are typically granted without too much problems. You might want to try the same thing in the US.
So I would guess that it is up to you to contact the FAA, provide evidence of your flight training and experience (since they formally have no clue whatsoever what an NPPL is) and see if they're willing to grant you something that allows you to fly solo or with passengers.
One of the things that the FAA will most definitely require (because they require it for PPLs as well) is a BFR. You don't need TSA clearance in this case either, since a BFR is recurrency training, not training for the grant of a new license.