How to get area type given lat/lon coordinates ?

This is exactly what "query feature" on the http://osm.org/ map does, using Overpass Turbo:

[timeout:5][out:json];is_in(lat,lon)->.a;way(pivot.a);out tags geom({{bbox}});relation(pivot.a);out tags bb;

Where you need to replace lon and lat with the coordinates of the point, and replace {{bbox}} by the current map bounding box (if you need that). This gives you the elements surrounding the given point. Note that this also returns administrative boundaries (county, state, etc.), not just physical elements - you'd need to filter on this, perhaps by throwing away boundary relations etc. See e.g. this query - note that the largest enclosing feature is the entire country.

Also, this will give you enclosing features - so for a highway (which is usually an unclosed way), you wouldn't get a usable result. For that, you may need the other query, "nearby features":

[timeout:5][out:json];(node(around:radius,lat,lon);way(around:radius,lat,lon));out tags geom({{bbox}});relation(around:radius,lat,lon);out geom({{bbox}});

Again, you need to replace lon,lat,{{bbox}}, and in addition radius (how far from the point to look, not sure of the unit) with your data. See e.g. this query - note that by setting the radius too high, there will be many objects; but with a radius too low, there may not be anything found. Depending on your goal, perhaps you might want to start out small and increase the radius in a few steps.

Pay attention also to the freshness data returned with results - Overpass uses its own cache, and updates it from the main db; this can sometimes take days, so newest edits may not be returned in the results immediately.