Find all higher-level administrative areas that contain a specified node?

You have already been told that re-inventing Nominatim will be a lot of work. I hope that you don't expect us to re-invent Nominatim for you on this platform ;)

First of all, streets are not usually represented by nodes, but by ways. You do not have "a node that has the tag street with the name Sukhumvit". You have a way with the tag highway and the name Sukhumvit, and that way will be built from nodes, which have a location.

Then, nodes don't have geographic boundaries so your question does not make sense. What you are looking for is administrative boundaries which may be given in the form of a single, closed way, or, more commonly, a multipolygon relation. Existing SQL database importers like osm2pgsql (used by Nominatim) or imposm will resolve these into proper polygons for you, as will the Osmium C++ library. You can also do this yourself but that would mean reinventing another type of wheel, as constructing polygons from the OSM data is not trivial.

Once you have these polygons, you can use geometric operations to find out which of them enclose your polygon. Again, PostGIS (used by Nominatim) can do this, and is indeed used by Nominatim to build such a hierarchy like the one you're looking for. Alternatively, you can use libraries like GEOS for C/C++ or its bindings in various scripting languages. Doing this efficiently will require building indexes.

But aside from the polygons, you will still want to look at single nodes that do not enclose the position you are interesed in, but are just in the vicinity - for example, there might be a node tagged "place=neighbourhood, name=Sukhumvit", which simply sits somewhere in the vicinity of the street that you are looking for, and you have to apply heuristics to determine whether or not your street should be counted as belonging to that neighbourhood. Again, this is something that Nominatim already does.

Are you sure you still want to build your own geocoder?