It's pretty much as simple as that - get the old version (which is a self-contained EXE that is not dependent on any external files, and can generally auto-download most missing styles - although authors should have been building non-standard styles into their NXP files anyway).
When you run the old version, you'll be given an Open File dialog from which you select the NXP file. Alternatively, you can drag and drop the NXP file onto the old version NeoLemmix.exe, or tell Windows to open NXP files with old-version NeoLemmix.exe when you double-click them - all three of these methods work equally well.
Some NXP files may require even older versions, but 10.13.18 (and all other NXP-supporting versions, for that matter) should be able to tell you which version is needed. I would estimate 80% of NXPs expect NL 10.13.18, with a further 15% or so expecting NL 1.43, as these are both "last versions before major culls" (1.43 the last before gimmicks were culled, 10.13.18 before radiation, slowfreeze, splatpads and some minor physics details eg. solid ceiling were culled).
Simon's post already stated this, but I will reiterate: No new content should be created for older versions; they are there for the sake of playing content that has not been (or cannot be) updated.